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This is a model S isn't it?
If memory serves there is a position on the front windscreen to place the fob when batteries are low or not working - base of passenger side windscreen wiper and then press door handle (might be passenger door handle?). Likely will have to try both sides if the instructions apply to US cars.
Failure of aerials can contribute to poor signals too .. so if usually car/phone is linked by home internet but you have good phone signal strength then try turning off phone wi-fi or turning off phone cell connection when trying the app?
A while back i did post a msg about changing my fob batteries for some I had lying around (new in packet) that failed to work - but it was fine when I bought a better brand - obviously the first lot had been sold as old stock and then sat at home for more months.
 
One other possibility might be the replacement battery itself is incompatible ... for real!

Aparently there is a new "childproof coating" applied to the likes of Duracell coin batteries intended to make them taste horrible to children so they won't be tempted to swallow them.
Apple AirTags have a problem with the coating and some simply don't work with a Duracell coin cells until you scrape some of the coating off or give up and use another brand that doesn't have any coating.

Other than that the spare battery used, if not brand new, might have been in storage too long and lost some voltage?
(My Volvo remote used a bigger coin cell that I'd kept a pack of as spares but 12 months later when the remote died it didn't like any of the pack :( )