My initial suggestion was anything that looks like a jazz bass with 5 strings and active, but the two you mention would be excellent choices. Not to say you can't use something more modern looking in many circumstances, but both the Stingray 5 and 55-02 have a classic look that will fit anywhere and can more than nail the tones you'll need.
Currently using a Fender P tuned EADG with a Hipshot D-Tuner. Wanting a Dingwall 5-string, or a Lakland 55-02.
On my Dimension V, I'm quite surprised that Ernie Ball Strings, the yellow-green pack starting at a 130 (I think) sound just fine. One my Precision V, I use DR Low Rider Nickels, same gauge and have been quite happy. Strings also depend upon your amp & cabs. I have an old school set up: Hartke LH1000 and LDS 2 x 10 cabs that are ported. BUT, it seems to work just fine.
I'd pick flats all the way because you get better decay and the tone " out of the box " is a good country sound. It's a little eye opening at first but it's perfect, really. I want to go to flats but half the stuff I play is in the modern vein and I need more sustain, kind of a bummer. I mean last night I was playing on my acoustic with flats and the more traditional stuff sounded better with the band than when we do our electric gigs. For those, I play an active j 5.
Kenny's guy is now a girl... Bio But back to the op, I have a Warwick thumb 5er that I use for 90% of my modern country gigs which has a male and female front. The other 10% I use a fender cs Jazz v and although it's more beautiful, the Warwick sound crushes the fender. I will keep trying to love that damn custom shop bass though. Eventually though, it'll just stay in its case.
I took your advice. At first I irrationally thought, "I love the tone of the rounds on my Jazz 5 for country music. How could anything be better?" But the tone of the flats kept calling to me. I had recorded my test of Sadowsky flats and during playback through studio IEMs, it surprised me how much massive presence came with those strings. And quite a bit of harmonic content, too. Fingerstyle plucking had a percussive attack veering toward what you'd get with a pick. And picking the flats was even more so. (Recollections of my prior experiences with flats were much "thuddier" than the Sadowsky's.) After three playbacks I said, "Hang it, I'm not saving those flats for a rainy day. Put 'em on now!" So glad I did.