Andrew,
First, it doesn't matter how good the team or the mixer are, if the instrument pickup or mic isn't doing a good job. I suspect you're using the Fishman C-100 pickup, one of the least pleasant sounding pickups for strings I've used.
There are a few recommendations I can make.
Fishman C-200 bridge:
C-200 Concert Series Cello Pickup - Fishman Transducers, Inc.
I haven't personally used this brand, but I have used the LR Baggs version on fiddle for many years.
I am loving this next pickup, and it can be removed and reinstalled by the player if needed. The Realist Acoustic Transducer for Cello:
David Gage String Instruments
Very nice sound.
Next, if you want to stay with a microphone, switch to a good clip on condenser. I recommend the Audio Technica ATM350, or the DPA 4011C for cello.
ATM350 Cardioid Condenser Clip-On Microphone || Audio-Technica US
DPA Microphones :: Close-miking a cello with condenser microphones
The ATM350 would clip right onto the side of the bridge. Don't worry about muting effect, it's minimal, and you make up for it with the close mic position anyway.
The last thing to recommend is to not use regular DI boxes, or even specialty ones like the LR Baggs-I can't answer to the one Chris recommends, I actually haven't tried that Radial model. If you must, use a very good DI like the Radial or Countryman. I have Countrymans, and they will produce an acceptable final sound for the bridge equipped fiddle.
What I really prefer and recommend is the ART tube preamps. The "original" or the MP/C:
ART Pro Audio
These two have higher input impedances than the other versions, and provide a better match to the pickups. The "tube sound"(there is a subjective factor involved too) only helps, IMO.
Best regards,
John