While I agree with the LW that they need to find some compromise, it's pretty clear from their letter that they've probably never owned pets. It's the quotes around "indoor only" that do it. No, if she's a responsible pet owner she can't put her indoor only cats outside for the weekend. They would have no idea how to survive and she'd be lucky to get them back alive.

That said, if she wants to keep seeing her niece she has to find some way to settle this. If her house is kept as clean as LW suggests, then possibly boarding the animals sometimes when her LW and her daughter are in town will be enough. Maybe sometimes the animals can be boarded and sometimes the LW can stay at a hotel, thus spreading the financial burden around a bit.

"just take allergy pills" isn't a long-term solution, but LW should probably acknowledge that her daughter is going to have to take them at least some of the time to cope as she's growing up. It's not possible or even necessarily desirable to always entirely avoid allergens like pet dander that can linger in the air.

Despite my having allergy-induced asthma, it never occurred to my parents to give me any kind of allergy meds on a regular basis, and growing up and discovering long-acting anti-histamines was a revelation for me.