Leica Rover and Tablet compatible with Trimble Business Center?

Is there any issue using this in conjunction with Trimble Business Center?

Yes, unless you plan on trying to export some commonly recognized format from the receivers and then import to TBC. Even then it will likely be a headache, as a lot of things tend to get lost in translation, especially when exporting from newer gear.

In my opinion it's probably more trouble than it is worth. Stay yellow or green, don't mix and match unless it's absolutely necessary...or you got a screaming deal and don't mind a lot of contortions and extra steps in your workflow.

Any other comments on these Leica units in general?

Dunno about these units, they are construction-side, which I have very little familiarity with. Leica tend to make solidly constructed and very reliable, precise gear. I'm not a huge fan of the field & office software (not super intuitive from my survey/geomatics perspective), but it is still some of the best you can get.

The other question to ask is, do you have good local support for this gear if you choose to buy and use it?

We mix and match a little data from time to time and I agree with Rover83. It's not bad with static, but I find the antenna modelings errors problematic. That's not helpful on vertical...

Antenna modeling should not be a problem as long as you get them from the same source, say NGS.  That has been the case for decades when dealing with post processing.

As Rover and Bionic have said, post processing is doable but bringing in RTK data for adjustments may be a headache.

Leica's construction gear uses its own software and may not work that well for a surveyor unless you are mostly doing construction.

Leica has a proprietary format for exporting raw data and does not play well with other manufacturers equipment.  I would suggest that you do not mix the two brands or you will find yourself beating your head against the wall.