Without actually seeing what the situation is, it's not possible to tell what it will take; but you're only dollars away in one way or another. A good repair tech should be able to do it in the least invasive way possible, and may likely be able to scope it before deciding how to approach the problem.
I typically don't use a snake except for tuning slides and the lead pipe -- depending on my flushing apparatus to clean out everything else. Gunk typically doesn't collect except at the bottoms of slides, and what does can be flushed if you do it right. Not worth the risk to me to screw something up or get stuck in order to clean a portion of tubing that almost certainly doesn't require internal scrubbing. Also, if you clean pretty regularly, you'll discover that snaking doesn't accomplish much because there isn't much there to accomplish. At least that's my experience.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)