Is there any way to figure out the phonotactics of a language and
create words that sound like they fit in, without having to learn an
insane amount of linguistics/"source language"?
There is: look it up. For example: Nahuatl Phonology.
Nahuatl phonotactics turn out to be pretty simple: maximally one onset consonant, a long vowel, and one coda consonant per syllable. Here's an Awkwords script that'll give you 100 random 3 syllable words conforming to Nahuatl phonology:
#awkwords version 1.2
If you don't care about the source language morphology, that oughta be good enough. Not all of the words will be winners, but you can generate tons and just pick the few you like best.
To get words that really look like plausible source-language words, though, you do need to care about morphology. In that case, you do have to learn something about the source language, but not necessarily "an insane amount". For example, a simple change to that Awkwords script will generate plausible dictionary-forms of Nahuatl nouns with 3-syllable bases and <-tl> or <-tli> suffixes:
#awkwords version 1.2
It is not too hard to look up lists of other morphemes and add their patterns into your word generator. If you want to get really fancy, you can try to find a pre-made computational morphological model for the source language (e.g., a PC-KIMMO or KLEENE or HFST file) and then generate random underlying forms to run through it, but that probably does start to get into the realm of "an insane amount of linguistics"...
There are a couple of decent middle ground, rather than copying the source language morphology. One is to generate a bunch of short, one-or-two-syllable con-morphemes of your own that match the source phonology, and then use those as the base units from which you generate conlang words, rather than going directly from the phonology.
Another is to try to imitate the source language phonemic distribution. For that, you basically just need a big list of source language words, from which you can extract statistical information. The simplest way to do that is to just count up occurrences of each phoneme, but if you know a little bit more statistics, or are willing to learn, you can construct a conditional distribution model for which phonemes occur most often in different particular positions, and then generate random words based on those specialized distributions, rather than the default uniform distribution. Awkwords can't handle that kind of model sophistication, but other word generators, like Logopoeist, can, and it's also not too hard to write up a custom generator just for your particular naming language.