I agree that the whole thing is very clunky, and done solely for the sake of more crossover, but the BW scene is very clearly meant to tie into Hawkeye.
We don't really know what Val's M.O. is as far as building her Thunderbolts team to oppose heroes. I know she's supposed to be a shady antagonist, but it seems like she's operating under some governmental organization. So it makes zero sense for Val to sic Yelena on Clint when he's a non-powered civilian. He's completely retired after Endgame and has been exonerated for his Sokovia Accords violations after he struck his deal with the government. What reason would Val have to put a hit on him other than getting several million dollars from a client?
Plus, like Slyonic said: Yelena would have immediately dealt with Kate in an at least non-lethal way unless she was being paid to not harm her. Otherwise, their drawn out cat and mouse banter is pretty ridiculous.
And I think the weather shouldn't be a factor at all. I've worked on plenty of timelines, and you constantly have to place scenes in the winter that have outdoor scenes with warm/neutral weather. In this very timeline we have Winter Solider taking place in DC and New Jersey during the month of January, but it's very clearly summer time, going by character clothing and set foliage.
