battery replacement for my Apple Watch series 2, 38mm

For repair & service:

watch Repair - Official Apple Support

watch Service Pricing - Apple Support

According to the above, Apple will replace a series 2 watch battery for $79.

Your best option may be to make an appointment at the Genius Bar in an Apple Store near you and let them look at it:

Genius Bar Reservation and Apple Support Options - Apple

I would not waste my money on that. Go for a new SE 2022