I don't play correspondence chess myself (if I have occasionally it's without engines), but I have written a few opening books. The research aspect of both activities share some similarities.
In the tournament, I'm assuming you're allowed to use engines and databases. I'd recommend getting the best engine available (Stockfish 12), downloading the updated TWIC database, getting the up to date ICCF database, and possibly buying any recent books on the opening in question. You could try looking into renting someone's CPU time on the cloud to run Leela, but this seems excessive.
Then, before the tournament the main thing to do is to understand general important patterns in the opening you'll be playing. For example, who usually benefits from certain types of positions that can arise, where do the pieces tend to go in various circumstances, etc. Also try to learn the plans for both sides (how to develop, where to attack, etc). This will be useful to help guide the engine: i.e., to know what moves to get it to spend more of its time analyzing, since if those moves correspond to core principles, they have a higher chance of being right. Although there are always exceptions to principles and patterns!
When you're guiding an engine, this means letting it think for a bit. Then, when it comes up with a move (ideally at depth 30 or higher if you're very serious, otherwise, depth 25ish is okay -- although if it's absolutely clear only one move is acceptable, no need to let the engine think this long), play that move, and let the engine think again. Once you do this enough times, you can see what the evaluation is for the position you reach a number of moves later. This is a much more efficient way of analyzing than just letting the engine think on the starting position for some indefinite amount of time. Also, don't just play out the line with the best suggested moves, look at other moves as well when guiding the engine, and map those out for a while. It's possible an engine's third best move is actually the best one, which you could realize after analyzing it for many moves deep.
The exception to choosing a move the engine gives when guiding it is if the game is still early and you clearly know the move goes against the principles of the opening (based off your study of it). Like say it's on the 8th move and the engine wants to go against widely accepted theory - there's a good chance you can just ignore it. Another indication that the engine's move is poor is if it conflicts with the most common practice from the ICCF database. In this case, even if the game is later on in its development (say move 20+), ICCF games spend days on each move and use engines themselves. Although keep in mind that Stockfish 12 might be better than a game in the ICCF database from a few years ago, since the engine has made significant improvements. As in some cases, it might be that Stockfish 12 can do better in a few minutes than a correspondence player could guiding an old version of Stockfish (or another engine) for a few days.
As for finding and testing new/unusual ideas, don't worry about coming up with these before the tournament. During your games, when you guide the engine you can do this. If you see some wacky move that your gut tells you is good, and if the engine at least doesn't laugh at completely right away, then go ahead and guide the engine through analyzing this move. The engine might be a bit skeptical of the move at first, but if you play through the best moves it suggests (especially for each move being given at depth 30+), you'll have a better idea of the new move's validity at some point.
