There are many types of coax.. Standard RG-6 should NEVER be coliled around a core less than six inches in diamater.. This is your first mistake.
They make some very small coax.. Very very very flexible.. But also very high loss.
Here is a thought... Your profile says DENVER, CO
Also in Denver, CO, is a store called Ham Radio Outlet
Large (VERY VERY LARGE) selection of coax and a good selection of folks who know more about it than I do (And that is saying a lot).
These folks live, and breath, Radio.. and COAX is .. Part of Radio.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times