Billy has been used to all the heat in Cali, so when moving to Hawkins, the summer wouldn't bother him all that much, right? He can take the heat. In fact, he LOVES it. He would try to surf in the hot sun back home all the time if he could. (stares at Neil)
And when the mindflayer possesses him to do its bidding its all while the flayer is in the cold damp basement of the warehouse. While the flayer is physically strong enough to yoink Hargrove (and explicitly done at night), its not strong enough to walk outside in the day, aka direct sunlight. would insta-burn that bitch to a crisp, which is also where the fireworks and sauna stuff comes into play. this disgusting creature cannot stand heat because heat will cook, will kill.
with the theory being: day = heat = fire = easy death for the flayer, regardless of whether the weather is cold, there's still the suns directness to contend with. and when the weather is patchy (spring/autumn), the suns temperature and direct rays fluctuate throughout the day, making it a little too risky for the flayer to time possessions/captures/kidnappings all while shambling back to the safety of the warehouse basement. this is where Billy starts being used.
The first time Billy is under control, he's confused and dizzy etc, and its because the mindflayer is adjusting to his body. It's like a virus in which Billy's body is attempting to fight the pathogens invading his system - but there are none because it's a possession, an overtaking of his conscious. theres nothing physical to fight so Billy adjusts the only way he knows how when something hurts him (STARES AT NEIL AGGRESSIVELY): forced acquiescence, numbness, disassociation. ....... the perfect meatsuit.
Billy when first at the pool develops bubbling heat rashes, excessive sweating, bright and confusing rays of sun blinding him - this all being from a puppet-esque version of the flayers hand inside Billy (the flayer pumping whatever it is into his stomach being the essence of the flayers mind control, etc etc, digression into the flayers possession being an allegory of SA, insert here).
but as the flayer continues to get stronger in the basement, it also adjusts rather AGGRESSIVELY quickly to how Billy's body works. like the venom symbiote meshing with eddie, billys systems adjust around the flayer's false permanence while the flayer adapts to the outside world safely using Billys physical being as a cover/coat.
another part being that: since its summer time, that means every human is outside in the open enjoying vacation time, pool time, schools are being let out. prime meat pickings right there. humans dont stay outside during the cold months, and the flayer knows this from personal experience, having seen endless human nature scenarios with vecna/henry. of course it would be the summer where they pick out of the pile with pure joyous malice - where they not only find the strongest physical body to carry their commander (billy) but also the rest of the BEST apt soldiers to form the physical MEAT of the flayer. flayer needs more skin and bones to create itself. why not use the apex predator of earth to completely overthrow it and create the hivemind?