New high hand prize--more or less payout?

The bar-poker league in which I deal and play changed the high-hand payout recently. I tried some very back-of-the-envelope calculations, and came to the conclusion that the new way will actually cost more, but I'm really not sure. Wondering if anyone has a better analysis.

High hand payout is:

Quad deuces, two in hand and two on the board. Must win the hand, but it does not have to go to showdown.

Old way paid $200 each win. New is a rolling jackpot, starting at $20, and going up $20 each night. (if there is a second winner in a night, it's reset and they get $20. It has happened, but probably not worth deviating from the baseline).

Now for some ideas on play:

We play tournament-style poker four nights/week. Generally ~40-50 players/night--so four or five 10-player tables. It's free bar poker, the play is generally quick. Since there's no money on the line, "tanking" doesn't happen much.

Of course, some dealers are faster than others. I'd estimate we get in about 20 hands/hour/table. Of course, the number of tables decreases as the night progresses. We usually go about 4-5 hours for the first game. I'd guesstimate we get out ~200 hands/night.

There is a second game in which the high hand is eligible. It's a quicker structure and smaller field. I'd guess that ends up adding ~75 hands. In neither case have I gathered actual data.

Part of the reason for the change was that it was felt (I think rightly so) that the jackpot was affecting play. People would stick around with deuces trying for the prize when they would've folded in a "normal" game. So, I guesstimated that quad deuces would happen somewhat less frequently (but probably not much, since the jackpot will get to the old $200 in just ten nights).

I believe the prize was paid 18 times in the last year. The interwebs tell me the chances of quads is 0.82%, but that's assuming a pocket pair. It also sez there's a 5.88% chance of getting a pocket pair. Having those data, I figure the chances of quad twos on any given hand is:

(5.88/13) * 0.82 = 0.37%

So, roughly four of every 1000 hands should pay. Let's move the hands/night estimate down to 250 for ease. That means it should've paid every night. Since that wasn't even close, something is clearly wrong with my thinking.

Any leads on how to gauge which approach is more costly? It doesn't matter from a business standpoint--it's not going back--so it doesn't have to be a detailed analysis. I'm just a curious now, and it feels as if I've overthought the problem..