Since apparently amplification isn't allowed, you may want to consider a particularly loud dulcimer. Some dulcimers are quite a bit louder than others. There are a lot of options depending on what kind of tone you prefer. In general, larger dulcimers that can support heavier strings will often be a lot louder than smaller instruments, however if you are looking for something portable, you may need to compromise. 

Here's a handy page concerning busking/playing/performance on NYC subways:
https://www.mta.info/agency/arts-design/music/subway-performance-rules
You'll have to obey their stated rules on amplification (since you mention 'pickups'). 

A hardshell case might be good to protect your dulcimer while moving around/crowded stairwells, rain, etc.

One cheap solution might be to buy or build a wooden "possum board" to put your cardboard dulcimers on while playing acoustically... such boards do help increase volume and resonance, without electronics. You could make one to fit your existing instruments pretty easily. Do a site search here for "possum board" and you'll find lots of info.

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I'm thinking about starting to busk in the NYC subways in the next few years (if/when I ever get the courage). Right now I have two Backyard Music dulcimers with cardboard soundboxes—one diatonic, one chromatic—but would love to acquire a more grown-up instrument with a built-in pickup that isn't TOO nice to the point where I'd worry about it getting broken in the subways. Do you have any recommendations? Should I instead get a guitar pickup for the cardboard dulcimers I already have and do my best with one of those?