Before the new entry fee was announced we had put plans in place to fly to Lima via BA. Our time in BA is less than 22 hours. Is there any way to avoid having to pay the $131 entry fee?

I am not sure if this helps you but if you are in transit you will not have to pay.

In some countries they might keep your passport if you are in transit and let you in the country but I do not know if this applies in Argentina.
I know that once my plane LAN from Montevideo arrived in the morning and I had to wait for the night flight to the USA, I could not get in Chile without paying. I opted for a free lunch offered by LAN and a good book and remained in the airport.

We do want to leave the airport and stay the night at a hotel in town.

If you go through passport control you will have to pay the fee. Only by staying in the transit area at the airport can you avoid the fee.