If it’s not hot, oozing, very painful, and smelly, it’s almost certainly not an infection. I’m not a doctor, but I don’t see anything that looks infected.
Tattoos take 4-8 weeks to heal. Enjoy your tattoo.
That's the way I beach it, too!
For sure sweat it out. I'm in SW Florida, and back in the 70's and 80's didn't bother with sunscreen. I'm paying for it now.