First of all, lottery tickets or other gambling losses won't help at all with the 1099 you get from APA. Gambling losses can only be used to offset gambling winnings, and those are reported on a 1099-G. If a pool tournament was considered gambling, it would be illegal to conduct them in most bars/rooms across the country.
Second of all, the 1099 you get has nothing to do with tax deductions made by APA from their revenue. Regardless of the reported value of what they gave you, they are only allowed to deduct the actual cost to them, as shown on receipts and invoices.
Third, the retail values of the prizes awarded are determined by the manufacturers of those prizes, not the APA. If the makers of the Outlaw cue put something special on it, like an APA logo or something, and say that makes it a "commemorative edition" cue, they can estimate its retail value at a higher number because it is scarce. Same thing applies to the jacket - even though I am an APA operator I cannot get one of the cues or jackets, so it doesn't surprise me to see numbers like that.
Fourth, if they could, APA would skip the whole 1099 business. As I mentioned, it has no impact whatsoever on their taxes. It's just something the government makes them do, so the government can get into your pocket a little more. They would get in big time trouble with the IRS if they didn't do it. Trust me, they would LOVE to skip the trouble/expense of all that reporting.
Fifth, the money you pay for league fees, minus the portion that is for stuff like trophies, patches, and other awards, is already reported to the IRS as revenue for the operator. For awards/money distributed locally, it is up to the operator to comply with IRS reporting requirements. In almost all cases, the local operator complies by awarding less than the IRS minimum. For example, if the operator awards travel assistance for nationals, the amount is usually less than $600 per person, so a team of eight can receive $4,799 without getting 1099's, AS LONG AS they don't receive more in the same calendar year for other stuff.
I believe you can deduct from your income certain expenses related to the singles tournament (such as travel and whatever entry fees you paid), but don't take my word for it. You should really consult with a tax professional for that.