I think the B3K at low drive does that far better than the VMT.
I use the B3K circuit all the time in my big band - low gain, low blend, tone at about 10 o' clock. Gives just enough bite to cut through without sounding bright.
Pedal wise - if you can find an old Zoom Power Drive PD-01 then you can get a lovely round tone on that. Gain to taste and treble down.
In fact there is one on ebay at the moment: https://www.ebay.co.uk/itm/zoom-power-drive-pd-01-rare-overdrive-pedal/192981858593?epid=1323321713&hash=item2cee9d7921:g:RQoAAOSwc2NdJFeh