Hiya @JudiciousSage!
First off, well done on setting a limit… Make sure you stick to it and keep an eye on it as it can be easy to blow straight past it…
Regarding your question, there are a number of places you can get un-farmable ascension materials:
- $$ Buy them in offers… An expensive but most reliable way of getting them
- Gem buy them from the shop… Most expensive way and not guaranteed to get an item unlike in 1 above.
- Titan drops. The higher loot tier you are, the more likely you are to get a rare ascension material. THAT BEING SAID, I have seen people get 4* items from loot tier VI and above so anything is possible
- Filling Chests. Number of chests you can get them from specifically monster, raid, war, titan and elementals
- Mystic vision. Least common occurrence but still possible.
You can also get them from summoning heaps at Atlantis… There are 9 (?) chests you can open each Atlantis. Each chest is unlocked by summoning 10 times (carry over from one summons to the next).
The easily most reliable way is to keep buying them but I recommend the Titan route. It’s cheaper in the long run 
To get a better chance, you should aim to get to a higher Loot Tier. Two links, first is for a titan data project I’m running which has some preliminary numbers on the chance of getting materials from titan loot tiers. Second is to a previous post explaining the various loot tiers.