Since it's a restriction, must have been put in place for your own good.
I buy CR2032 from random brick and mortar supermarket chains (e.g. in EU, from LIDL). The cell brand doesn't seem to matter much if you put them to use immediately. However, the shelf life is better for brand names CR2032, brand names can still be good to use after 5-10 years in storage, while a no name will develop high internal resistance in 2-3 years and become unusable.
Another thing that might help, if there is enough room inside the appliance, often the CR2032 button (no more than 3.3V when brand new) can be directly replaced by a rechargeable Li-Ion cell (3.7V avg, or 4.3V top most when fully charged, but you don't have to charge it fully, it is not much energy stored over 3.8V anyway). At first I was worried the 4.2V might fry the appliance, but turned out I was lucky so far and didn't damage anything yet.
Did such CR2032 -> Li-Ion replacement already for a body-weighting scale (more than a year ago, time to measure the voltage again, calculations tell one recharge will last for 10-20 years
https://www.eevblog.com/forum/beginners/li-ion-cell-instead-of-a-cr2032-for-a-domestic-wight-scale/msg5339312/#msg5339312 ), and for a 7 segments mains clock which has had a CR2032 backup for its RTC.
The Li-Ion rechargeable are from former mobile phones, the battery will still work for decades after they get bulged, and no matter how much capacity they lost, they will still have 3-5 times more than a CR2032. And no, old bulged Li-Ion from former mobile phones do not suddenly ignite, or explode in flames simply because they are bulged.