Trying to get value with turned quads

Hi all. I apologize for some missing details, I am a fairly new player and still learning to catch everything. Trying to figure out how I should have played this.

$1/$2 NLHE
Large casino, very late night, probably around 3-4 AM
Action was very slow, limping common, check-folding common, not many large pots at that hour
Generally a very casual feeling
Multiple people at my table had complained about getting dealt a premium hand but getting no action at all
Tablemates were a mix of mid to older recreationals, one mid-30s semi or aspiring pro, one early-20s recreational
Hero: 35 y/o recreational, deep stack, had been trying for TAG earlier in the evening but loosened later, I was participating in the limp/check-fold atmosphere at the table

Hero: BB, dealt A2o

3 (I think) people limped in, button raised to 5
I had seen the button overvalue hands, chase draws, etc
I called, so did 2 limpers

(So far, this was not an uncommon way for pre-flop action to go at the table. I didn't consider my hand great, but with how loose the table was and with the villain's behavior, I decided it was good enough to call, see the flop, and evaluate there based on whether I improved and if there was any aggression.)

Flop was x22
I don't remember the other card, believe it was mid-rank, 7, 8, 9 or so.
I checked, it checked through

(I had originally been ready to fold at any aggression since I didn't consider my hand all that strong. But I was very happy with what I assumed to be well-hidden trips. My plan at that point was to check to to pre-flop aggressor and call if they bet, bet on the turn if not.)

Turn was the last 2
I wound up checking, and it checked through again.

(I was not ready for this and had not considered at all how to react to a hand this strong. I basically froze like a deer in the headlights. Amateur, I know, but this was the first time I had ever made quads.)

Flop was a blank, I will confess I don't remember what it was exactly. I know it did not pair with the first, unknown flop card.

One thing to mention, the casino had a High Hand Jackpot that ran every 20 minutes at that time. Whoever had the highest had during each 20 minute period got a bonus $100. Quad 2s would be a very strong contender, especially late night when there weren't as many tables running. However, the screen showing the current winning hand and the countdown timer was situated behind me. I felt turning to look at it would be extremely obvious.

There is also an additional stipulation that the pot must be $30 minimum to qualify for the jackpot. And to clarify, that does mean a bet would have to be called to be added to the pot. When the river came, the pot was not big enough. I felt like I absolutely had to get money in, both for a general desire to get value from my hand and to qualify for the jackpot. The action had checked through twice at this point and it seemed very likely that nobody connected with the board in any way.

What would you do in this situation?