You would do just fine with your Tikka idea. There are other companies nowadays that also make some good stocks, so do some research on that. Aftermarkets parts for Tikkas has exploded in the last few years, so no worries there.
I've built 3 customs, and the two that were built off Rem 700 actions were around 3k. My last build that was a Defiance action, carbon barrel, was around $4k-4200. I'm not sure I would go the custom route again. It's a pain to source parts, I always wished I had done something different, and re-sale on them seems to tank pretty quick.
Seekins Havak Element for $3k is the easy button. Scheels has them, as well as a number of other higher end production rifles you could check out and research.
FWIW, I started my LR experience with a factory Rem 700 SPS Varmint, swapped out the stock and trigger, and got to shooting.