I busked in the Toronto subway stations for years(fiddle).Amplification was allowed but I was a purist-so much so that I needed physio every couple years to alleviate the damage from playing too hard in order to be heard above the din.One quickly finds out which are the quieter station spots(corridors away from the trains) that may also have a nice echo.Fiddle projects way more than the dulcimer,both sound wise and appearance wise.I would advise a pickup and a lightweight amp you can carry in a knapsack if amplification is allowed.In Vancouver I finally bit the bullet...using backing tracks I made myself played through a small amp which I could also plug the fiddle into.It became much easier and I tripled my money. There were so few paying jobs for me at least in Vancouver that I had to do something to survive..there went purity but I had a ball making the backing tracks....Schatten in Kitchener Ontario makes a really good contact dulcimer pickup.  and there are many cheap good quality small amps that run on batteries.P.S...I dubbed all of the backing tracks onto a Mini Disc player which I then plugged into the amp.Perfect small technology.

I think the loudest acoustic dulcimer I have is my Tennessee Music Box (TMB). It is home-built, and I described it in the thread

https://fotmd.com/strumelia/group/42/box-dulcimers

With the 1/2 inch plywood faces it is substantial enough to survive a lot of rough treatment. It might be used as a shield It has a screen door handle on one side so it doesn't need a case

There are very few dulcimers available with proper pickups installed for electric amplification. The piezo types - either built in or attached - will pick up both any bump on the dulcimer or sometimes other sounds not associated with the instrument such as passing trains.

John Knopf built a hybrid TMB about a decade ago. If you are buying, not building your own, at least exchange messages with him.

https://fotmd.com/strumelia/group_discuss/1018/electric-hybrid-tmb

Amplification is allowed if I get a license for it (which I assume many of the musicians who busk in the subways must be doing, there are tons of people with speakers!). Thank you both for your suggestions—I may look into a louder dulcimer.

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?