What DI are you using? Do you use drive pedals? Using a big valve head means a fairly important chunk of your tone is fromt he amp and speakers, and the DI is before them. Totally not worth lugging a Mesa about if you are DIing before it. If you are using drive pedals, then a DI, that can also be a problem, they produce lots of nasty highs the speakers in a bass cab won't reproduce, but will come out of the PA, meaning the treble has to be cut, which might be the muddyness to your sound.
Mic up, and give angry looks to a sound man who refuses, or get your own bass specific DI, plenty of Sansamps going.