“Allowed” and “Recommended” are not the same thing. I would never do it. But as long as you’re aware of the potential for issues and willing to deal with them, including making sure this is clearly stated in all pre-tournament publicity, it’s your event.
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You could pair yourself in and out of the tournament, round by round, to keep the number of players even, round by round.
You could also do this with a house player – pair them in and out of the tournament, round by round.
Either way, you could give the extra player (whether it’s you or a house player) a pairing score, round by round. This pairing score could be an unplayed game (U), a half-point bye (H), or a full-point bye (B). There is no reason why the extra player has to be paired against the player who otherwise would get a bye.
If the extra player is high-rated, some (or all) of their pairing scores could be a full-point bye. If low-rated, their pairing scores could be mostly zero-point byes. If rated somewhere near the middle, they could be given a lot of half-point byes.
A policy like this tends to be a good way to avoid hard feelings, since every player who entered the tournament “normally” would get a game in every round.
Bill Smythe
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For a while I had a former member who would hang around at events anyway, just to watch others play, and he was my ‘designated house player’ for several years, he’d play if there were an odd number in a round.
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Granted we have a huge membership (300+) and have anywhere from 35-50 players show up for the weekly club meeting, but we solve this simply by sharing the pain through a club rule. So granted, some of what follows simply works because of the scale of the club, but hope this helps you choose a way forward …
Our weekly club games are on a “Ladder” (list of current members, ranked by rating) where, like in a Quad, you play someone close to your rating (but one game instead of three). Usually players are paired with someone within 50 rating points.
Results are submitted at the end of each month, ratings are updated at the same time and players rise/fall on the Ladder as their ratings change monthly (in traditional tennis or squash ladders, challengers who win swap positions with the defeated).
We pass around the bye to the lowest players by protecting them from getting it again too soon. With such a huge membership, it can be 9-12 months before someone gets a repeat bye.
We mark who has had the bye previously with a notation ( + ). We clear out the byes periodically, so folks eventually get put back in the rotation. New members are “protected” for their first few meetings from getting the bye immediately.
Color is White for the lowest rated in the pairing for your first game against an opponent, thereafter you switch colors every time you play that opponent. If you forget last color, you shake-and-choose pawns. Flip side concern to lowest bye … top rated player who are new to the club (or recently increased rating) can often get Black in such a system, and it can be MANY games before they get White … so to keep them enthused, we allow that top board to shake-and-choose pawns for their first game against an opponent (actually a recent rule). I suppose you could always shake-and-choose for every game … but this is tradition going back to the 50s.
There’s also a rule that no one is forced to play the same person twice in a row (we track last opponents), because if your rating doesn’t change much you end up playing the same folks over and over. This way you at least get a bit of a break.
We also have two other club Ladder rules, by agreement of both players, you can … pair yourselves … and/or play a slightly quicker time control (we have a long regular tourney time control). These aren’t often used.
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Why should low-rated players be the only ones stuck with byes? The high rated players get all the perks!
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I see three potential solutions.
Option 1: Add yourself to the tournament with zero point byes for every round. If you pair and there’s a someone assigned a one point bye, then delete that round’s pairings, remove your zero point bye and then re-pair the round.
Option 2: Pair the round normally, ask any player who gets assigned a one point bye if they want to play an extra rated game.
Option 3: Get the guy with the recurring one point bye to recruit a friend and buy them a US Chess membership to be a house player and pair them the same way you would pair yourself in Option 1 or 2.
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This is what I did. I offered to play each rounds bye player an extra rated game. This gave them their bye as a full point, and got them the rated games.
Don’t know if I can do that all the time, though, I am only rated about 1550, so the extra rated game didn’t lose the low rated bye players any points, but in my distracted state as the TD, I actually lost one game to a 900 player! Cost me 40 rating points for trying to be a nice guy! 
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I detest being a playing TD. I too am rated about 1500, but play around 1100 (or worse) when also performing as a TD.
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Did you say that you were not finding a place to hold a tournament? Where I am, we found a building used by card players, The Bridge Center. The rent cost is only $5 per player, but perhaps that is because one of the chess players also plays bridge there. If you have money, you might rent a room or two at the YWCA. You may have to come with your own insurance coverage though.
I hope to help run a tournament in September where I may play as a house player if there is an odd number who joins the tournament in the beginning. I won’t try to win anything. I will get all the byes that are to be had unless someone asks for a bye.
The thing is, should I give all the people who have byes, zero point byes? As far as my byes, perhaps they should be 0. What about other players’ byes? They should not get a free point toward the money if they don’t actually play a game. Zero point byes may work for them too. But I don’t know. Should they get 1/2 point byes?
The most common rule is to give a player requesting a bye a half point bye. Some TDs limit the total number of half-point byes that can be requested to 1 or possibly 2 in a longer event (say, 9 rounds.) Many events require that last round byes be requested and posted early, such as by the end of the first round. Also, once a last round half-point bye has been approved, the player generally cannot change their mind and ask to be paired in that round.
In larger events it might be reasonable to give players early in the event a full-point, half-point or zero-point by depending on where they are in the field. It doesn’t make much sense to give the top players a zero or half-point bye in round 1, it gives them a pairing advantage that may last for the rest of the tournament. The reverse is true for the lowest-rated players.
With pairing programs you can simulate events and see what various rules on byes might do to the final standings.
One mall we used to hold chess tournaments at frequently started requiring a $1 million liability policy, that got pretty expensive. And then they stopped renting out that space for most types of events, including chess tournaments. Another motel with decent affordable meeting space for events up to about 200 players kicked us out because we were too loud, too messy, wrote on their tablecloths, and didn’t spend money in their restaurant.
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