Off the cough suggestions, purely subjective
While when possible I'd tend to prefer either no sound reinforcement or mic'ing a nylon string guitar that is not always practical . . . Last generation of piezo's seem to have some improvement over earlier efforts I've had reasonably good luck with K & K pickups [here] They come with a variety of options. I currently have an entry level installed on a nylon string guitar (would not characterize what I do with it as 'classical') and for what I need it to do I am OK with it. Have several other systems installed on steel string (which is what got me to try one on the nylon) and am quite please with piezo/mic combo on a D28 . . . Live it pretty much sounds like a guitar. Just to test used it on a session and while that will not be standard choice (tend to prefer to mic acoustic guitars) it worked on that session
The small pre amps K & K offers are probably worth the money . . . Subtle, useful adjustments to tone are under the hood, small screw driver pots but it tends to be worth the time they are not quite set once and forget by and large you are adjusting those for guitar not venue
I would tend to recommend running through some sort of mixer, prior to amp/speaker. Increases gear to lug, complexity of set, potential for problems but over the years I've found them to be more valuable then detrimental (even for 'just' instrument and certainly for instrument + vocals). I've used 2-4 mic pre models from Mackie (still have an original 20 yr. Old 1202 that still occasionally sees service), Spirit discontinued folio, Yamaha, maybe Carvin and probably a bunch of others. All have strengths and limitations. You might luck out with Behringer but like a lot of people I would not recommend their gear for anything that requires transport and is mission critical . . . Shorting out phantom power not something you need to experience.
In somewhat similar situations, including tight budget, I've use AKG C1100s and a another back electret SDC audix (whose number I might never have known, I just think of the Audix that surpassed me by how well it worked live mic'ng mandolin (on the first show for which I used it I thought I was going to be using an upscale CAD VSM (also long since discontinued) . . . For what ever reason the audix worked better . . . Bought it from the tech before I left the stage. Main point is that while neither of these, generally speaking, would be 'goto' recommendations for me they have and I would expect them to continue to work in the type of situations you describe. You need to audition a bunch of mic's until you find the ones that consistently work for you.
I've also used (use) one of the AT lavaliere mics attached to the instrument. Worked better on Mando then nylon string (this time deployed more or less as 'classical') But in a similar attachment arrangement I've used Naiant [here] mics, but again the ones I currently used have since been discontinued. But I've used both the cardiod and omni's attached directly to instrument with some success
I've been pretty OK with the JBL powered EONs for close to a decade (yet again I'm using product supplanted by newer models) for what I'm assuming to size of venues you're addressing the 10's would be fine. Even for standard singer/songwriter coffee house type shows I tend to use a pair plus a sub (not for thump but at low volume for 'detail') but even using a cross over that shunted the below 100 Hz stuff to the sub had a bass player comment on how well the 10's present the bass (probably because the below 100 Hz stuff was routed somewhere else) . . . But main point was even after a decade he was favorable impressed by the JBL EON 10's . . . He has one of the electric stand up's that he did not typically use with me but after hearing the 10's (interestingly it was not the first time I'd used them with him) at one show brought that (in addition to standard electric bass) to next show and we just DI'd it straight to board to PA Seemed to work OK
Standard disclaimer is I have no financial connection with any product mentioned, but not only personal experience with specific gear but with the one noted exception mention brand names not out of brand loyalty but because my dealings with those mnfc product has had enough success that I would at least consider stuff with their brand (standard caveat is that I'm sure each mentioned manfc markets absolutely execrable product . . . But, generally speaking I'd still tend to trust JBL more the Dedringer . . . As always mileage will vary)
Recap: if I had to do type of show you suggest tomorrow: probably use guitar with internal piezo, plus add the AT lavaliere, K& K pre amp and AT are both 'XML' compatible so would not need DI, use a mixer I'd probably throw in a 31 band graphic, which depending on room I might or might not even attach and might use it as channel insert on one input or on mixer's master bus (if there is a problem with the room channel EQ tends to not be sufficiently narrow to address effectively) then mixer to powered EON 10's (probably w/o the sub)
