Honeysuckle can have so many stems that cutting and treating the stump is difficult. I've heard of good success using a backpack sprayer in the fall while still green (and maybe a step stool for us short guys :grin
I've contemplated mowing it, waiting for regrowth and then nuking it with roundup as well, but never tried it.... man what a guy could do with a good forestry mower!
Big picture... bush honeysuckle is worse than ironwood. Ironwood is undesirable no doubt, but it is native and can be controlled with proper treatments... often times a single cutting combined with some increased light through the canopy will severely limit its impact in a woodland. Ironwood is really just another step in the successional ladder of a forest.... succession can be manipulated.
Honeysuckle, on the other hand, is non-native, grows fast in the shade and sunlight (unlike ironwood), and is adaptable to really wide range of sites. In my opinion, if you have bad honeysuckle issues, it is important to get it controlled before doing any other TSI work because once it gets sun, it will continue to flourish (and spread terrible fast) and like stated previously, is very difficult to control.
Honeysuckle was one of those stupid things we (people) can blame ourselves for, but it might be the biggest contributor to the 'stick to natives' argument.