I agree with Ken: you want an amp with the best 'clean' channel that you can afford, with the power range that you need. You need neither wailing blues 'tone' nor stadium-deafening power. Check out the amps from Roland.
My banjo has a BarcusBerry stick on transducer pickup installed under the head. What amp would work best? a dedicated Acoustic amp with a built in pre amp? A tube amp, maybe a Fender Blues Jr?
It depends on what your desired use is. If it's sitting around home and for your personal pleasure I can recommend A Roland Street Cube, it has very nice built-in reverb and Roland's outstanding chorus effect, all in stereo. It's also good for busking and smaller venue use and you can busk for hours using a set of AA batteries.
Larger venues might require more horsepower, although most anywhere that requires more power will often have their own PA or will require the band or performer to have their own. In either case your amp would serve as stage monitor for you and the PA handles the front of house and stage monitor needs.
Without knowing how it will be used it's difficult to make a "What amp is best" recommendation.
The important thing to take into account is no matter what amplifier you choose it's never going to sound any better than your pickup. That's why you'll often hear it recommended to simply use a good mic. Piezo transducers of any type generally have a deserved bad reputation, so temper your decision with that in mind. Part of their poor performance comes as a result of the physical characteristics of the material. They produce a strong output coupled with a high self-impedance; those characteristics coupled with the capacitance of a standard cable connection between the instrument and amp form a RC filter network that degrades the sound substantially. That can be somewhat ameliorated by using a short cable (or special low capacitance cable) coupled to a high impedance input, typically 1 megohm or higher. That input can often be found on good "acoustic" combo amps. Check the specifications of any amplifier you might consider.
My Barcus Berry never sounded good. I tried three different Barcus Berry branded pre-amps that I had bought in the 70s, 80s and then early 2000s. All junk. All were high impedance output through quarter-inch cable. I gave them to a friend who wanted to tinker with electronics. I think I actually threw the pickup away once I swtiched to a Pickup the World film transducer. I bought an LR Baggs Para Acoustic preamp and am very happy with the sound I get through PA. It can output low-impedance through XLR cable and run on phantom power.
But the sound is not as natural as through a good microphone.
If my band becomes active again, I'm going to invest in a new preamp/DI box called a Tonedexter. It learns how your instrument sounds through a microphone and then adjusts the pickup sound to match. The videos are amazing. $400.
Tube pre-amp can possibly add some compression, warm up the sound... but electric guitar amps are designed for electric guitars (not acoustic instruments).
What you might want is something that can serve as a personal monitor; designed for acoustic instruments, mic/PU; like a Fishman Loudbox; has DIs out to main board, as well can have a monitor feed (from board) in; make personal balance.
"...it's never going to sound any better than your pickup. That's why you'll often hear it recommended to simply use a good mic."
Just to add an observation; many folks think they "need" a pickup... (for banjo, fiddle, mando, guitar...) because of hearing others bad stories, or recalling issues with feedback (perhaps due to poor set-up or use).
A properly set-up good microphone (and sound system) would actually work fine if not better for most folks in most venues; can often achieve plenty of GBF.
What some folks might need/benefit is just better understand of sound system. First is understanding mic; polar patterns (over different frequency ranges); distance; and it's placement in relation to monitors (and reflections). Often the GBF problem is just really poor placement.
Additionally, sometimes the problem is due trying to make the stage too loud. (Long term it is not good for ears). The problem is often more one of unbalanced monitor mix, not taking time to do a good sound check.
Just to note... piezo's alone don't always do much better at solving GBF, with poor setup; as noticeable with acoustic guitars;... banjo heads can act as a big mic diaphragm. (that is the monitor speaker sound is picked up by the instrument head/body, retransmitted)