CTR by far

I'm gonna jump in here and say damage boost more than either.

The whole idea with autogun charges is that they're slightly more dangerous to use but do massive burst damage.

Why would you need ASI when everything in front of you is dead from the charge you just shot?

That's highly incorrect. Switching doesn't cancel any form of frames, that's what shield cancelling does. And as for shield cancelling, it speeds up animations by a percentage rather than a set amount of frames which means that it's much more obvious and helpful on slow weapons such as Autoguns and Magnuses as opposed to fast ones like Antiguas and Blasters.

I agree, they're both useful but CTR more than ASI. For many weapons, it's probably best to choose whichever bonus you have the least of, since experienced players say the marginal value decreases with more boost, but Autoguns benefit more from CTR, and by enough that it might make sense to max out CTR even with no ASI boost. It's not as extreme as with bombs, though.

If you're wondering which trinket(s) to buy with Autoguns in mind and/or which Sprite Perk to equip with them, CTR is better than ASI if you don't have either at max, but I don't know whether the gun damage options are more or less useful. Clearing a room with one charged attack is a nice ability to have, but it might take a lot of damage boost to make the difference between being unable and being able to do that. Also for non-UV boosts, which other weapon(s) do you use with Autoguns, that can share those boosts? For example, I have long felt that Autoguns and Alchemers are just asking to be used together, but since the big update I'm also warming up to Blasters with Autoguns and Magnus with either Autos or Alchs.

For UVs, the better of CTR and ASI might just be whichever you first get a big one of, because whichever UV you end up with you can still configure for CTR Max plus some other boosts, without any status penalties, and that's not even with trinkets. Given the usefulness of heating our weapons and the cost of targeted UVs, CTR Med should be considered standard. Even if you eschew Chaos for status effect reasons, the Seerus Mask and full heat give +4 CTR against a max of +6, and the last two can easily be had from a Sprite Perk unless you try really hard to neglect your Sprite. That's before the armor bottom, which should give you something as well.

How do you know that the ASI influences only the time between the bursts? I'm honestly asking, because I'm always on the lookout for new information.

The general opinion is that ASI influences various aspects of the attack animations. In particular, ASI does affect charge attack rates, very slightly. (See offhand remark here, for example.) So it can't just affect the interval between regular attacks.

But I do agree with you, that CTR is generally more useful on Blitz Needle.