Is it possible for a dialect to be agglutinative but for the “base” language not to be?

Yes, my fellow respondents have rightly raised the definitional issues that would give one pause about agglutinativity.

I’m going to be less scrupulous.

The difference between fusional, isolating and agglutinative languages is a significant typological difference—although of course, as with anything typological, there are shades of grey that it ignores, and square pegs that it seeks to stuff into round holes.

Languages change their typology over time. It is however not expected that two dialect would diverge significantly, through internal forces, into say a clearly agglutinative and a clearly fusional version. If their grammars had evolved to be that distinct, you would expect them no longer to be considered dialects of the same language.

However: intense influence from other languages can accelerate that kind of typological divergence in a dialect. Cappadocian Greek, to bring up the only example I can think of right now, was under intense influence from Turkish, and in fact was in the early stages of language death. The most Turkicised, southern dialects (e.g. Ulağaç) ended up having vowel harmony, and some of its inflection was starting to look agglutinative (although I fear I don’t remember details).