Boss boosted by the weather?

If the move's type matches the weather boost, then yes. For example, all of Groudon's charge moves are boosted in sunny weather.

The key, though, is that if the boss's type is boosted, you catch a level 25 of that pokemon after the raid, not a level 20. Examples:

You fight Tyranitar with Bite/Crunch on a foggy day. Since fog boosts Dark (and Ghost), the boss's attacks are +20% damage. And since Ttar is Dark, you can catch a level 25 Ttar, not level 20.

You fight Tyranitar with Bite/Crunch on a partly cloudy day. Neither Bite nor Crunch are boosted by the weather. But you get the chance to catch a level 25 Ttar, because partly cloudy boosts Rock.
(This is the good option for you, getting something for no extra cost.)

You fight Tyranitar with Iron Tail/Flame Blast on a Sunny or clear day. The Flame Blast attack is boosted, since clear boosts Fire attacks. Iron Tail (steel) is not boosted, since it gets boosted by snow. And the Ttar, being Dark/Rock, is not boosted, so you get a shot at catching a level 20 Ttar, not a level 25 (since neither Dark nor Rock is boosted by that weather type). This is the bad option for you, harder raid and no benefit.