Metal String, Solid Body Electric U-Bass for a Reasonable Price?

You mean something like this?


It's not a U-bass; it's a Fame M-03 Mini Bass: 20.5" scale, 38mm nut, 19mm string spacing at the bridge, tuned ADGC.

 

The rubber stringed basses do use a piezo pickup.

Using standard steel strings on a ~20" scale bass, you're not going to be able use standard tuning. The strings will be too loose, probably so floppy they'd barely even function at all. You'd have to tune quite a bit higher, like Volker Kirstein mentions above.

The rubber strings, originally seen on the Ashbory bass, were developed specifically to achieve standard tuning at such a short scale.

 

The roundwound strings for ukulele-scale basses aren't really "steel" strings - they have a nylon core, so they won't work with magnetic pickups.

 

Rondo Music Bass Guitars

Even if they come with rubber strings, you can swap them out for round or flat wound nylons.

 

Per the above answers:
- if metal-on-nylon (basically scaled-up classical guitar "low" strings) is included, some uke bass models come with that kind of strings stock, but it really shouldn't matter - you can simply buy the nyloncore roundwounds separately and put them on the ukulele bass of your choice. Please note thay aren't sticky nor overly stretchy, but they do roll under the fingers;
- if we're talking steel-core strings only, instruments tuned standard EADG tend not to be shorter than guitar 7/8 scale (the 23.5" of the Orangewood Oliver Jr. Bass Live is the shortest I'm aware of, but I've seen reports on the BeaverCreek Travel Size Bass being 22"). I think TBer @kohanmike can make steel-on-steel strings work on a few of his 22+" basses (they're flatwounds though, ergo not particularly bright).
You could go even shorter, but you'd have to pick your poison (tubby and chorusy tone due to stiffness of strings too fat for the scale; or unacceptably floppy, sproingy thin strings; or tuning higher than standard).
To be thorough, steel-core strings for shorter scale, namely 20-odd-inch ukulele basses, have been attempted (the 508/BT set by Pyramid, apparently still available here, and similar sets by La Bella that didn't go beyond the small batch phase), but they were nylon tapewound and didn't have "bright" as a goal.
(They also didn't work, intonation-wise, as drop-in replacements on fretted uke basses because they would need the bridge moved back for proper compensation, but that's another matter.)