The post from pdxjb above (post
#368) is about the best, most current summary I've seen.
Any problem you run into carrying on your vape kit would most assuredly be the juice bottles, and perhaps the tanks, if they are filled.
All liquids must comply with the TAS's "3-1-1" rule and while TSA agents often have different interpretations of their own rules, any random agent could easily consider the filled tanks as liquids.
3-1-1 for carry-ons. Liquids, gels, aerosols, creams and pastes must be 3.4 ounces (100ml) or less per container; must be in 1 quart-sized, clear, plastic, zip-top bag; 1 bag per passenger placed in screening bin. The bag limits the total liquid volume each traveler can bring.
Source
I always put juice bottles and filled tanks in my zip-lock baggie, just in case. If they won't fit, I leave them at home.
If it were me, I'd take less gear to ensure the liquids fit the baggie and if I really had to have 17 bottles and 13 tanks, I'd ship them to my destination (and then home) or pack them in a checked bag.
This is based on my TSA experiences. Your mileage may vary.
Good luck!
ETA:
By the way, that's a great kit you've put together!