I’d do the barn and an outdoor.
There are times in the winter where I can’t ride due to weather but a really nice outdoor does negate most of that. Plus, I’m of the mind that riding only 3-4 days/week from Dec-March is probably a good physical and mental break for my horses.
Getting both for $100k is unfortunately probably not going to happen. You can get a decent shell for $50k and put in the guts yourself if you have the skills and tools. Otherwise I’d say you’re looking at a budget barn for $70k and an outdoor for $30k if you’re lucky.
FWIW:
We put in an outdoor for about $8,000-10,000 over the course of 2019 and 2020 but did all of the grading and work ourselves.
Put up a 30x40 pole barn last year for $12,000, one sliding door, one walk door and two windows. No electric or plumbing or concrete. Also not including any ground work or excavation. Literally just the building.
In the Midwest where labor is cheaper.