Light + Water = algae? That's what could cause it. Not too complicated. It's a good sign and normal. I know people who put bettas straight from the fish store to a container with water (plus tap water conditioner) and they survive just fine without any cycling. That's not to say to do it, it's just to say you should be safe. I'm sure you're in a hurry as I've never seen anyone measure water parameters twice a day, but it's like a boiling pot - you watch it, it never boils. If you want to be safe, keep what you're doing for another few days so it rounds out to 2 weeks total and then you should be good.
Now I can't say for sure if 6.0 is okay for bettas, that sounds kind of low. From some research I've done, they look like they're better at more neutral 7.0 range and lower. The real problem is just PH swings for fish as if it's constantly 6.0 then your betta could adapt but I'm not an expert on them so I can't comment further. Definitely don't add any PH UP or additive like that. What's in your tank? Anything to lower PH, or is it just tap water and it's naturally 6.0 where you're at?
It is a betta and I'm sure they'd survive your tank and 6.0 water as is right now, but since you obviously care (which I commend you for as many bettas don't get very good homes) then i hope someone with more knowledge on them can answer as I've never had a betta before.