Some companies will definitely allow you to do this! What you would have to do is call a CSR for whatever card company you're looking for the increase, ask them some preliminary questions (e.g. WIll this be a soft pull? Can you pull from a bureau of my choice?) If the answers satisfy you, and you can confirm from the CSR that yes, they will pull from TU or EX or EQ, then go for it!
In terms of making sure they pulled the right one, you'd need a credit monitoring service from whatever report. TU and EQ can be easily obtained, EX is a little tricky. You can find free trials or free services though that'll allow you to review recent inqs.
Best of luck!
Asking a front line CSR about a soft pull/hard pull and which specific CRA is going to be pulled is an excersize in futility.
You would need to speak directly to an underwriter to guarantee the results you want. Even then, policy differs greatly per bank, and they may be suspicious that you want a certain CRA to be pulled. Unless you have derogatories on certain CRA's, hard pulls are not that harmful.
If I'm eligible for a soft pull for a CLI, can I really request which CRA to pull from? I had heard you could so this... What's the best way to make sure they pull the right one?
You can request, but the general conception is that you have something to hide. Most reps will shut you down thinking there is something you don't want them to see.