Roundup is okay,(the industrial stuff) You need a card in my state to buy it. There is some other herbicides that are better for woody species that will work better. Be careful of water run off. If you really want to kill the multiflora rose, mix roundup with diesal fuel as the surfactant, then come along and bush hog it.
Another option is to rent a skid steer with a bush hog and only cut sections in order to get the deer to move where you want them. They will take the easiest route. I agree with the other guys, if burning is not an option. If you spray then burn would be the best option in my opinion.

Honeysuckle is the first plant in the spring to green up and the last leaves in the fall to turn and fall.  Conseciently, it is a very wet plant and doesn't burn well.  Even after uprooting and leaving in piles for a winter it take quite a bit of accelerent to consume the brush.  I know, been there, done that.  
Because if it is cut but not instantly sprayed before the wound skins up it will actually grow even more thick and wooley.  As mentioned before the best way to rid an area of honeysuckle is to uproot it.  It is not difficult when small but it helps to have the proper tools once it bushes out into multiple trunks.  A skidloader is optimum, considering you have one and the room to move it in the woods.  Another option is this:

http://www.misterhoneysuckle.com/

This is a good tool for those with limited heavy equipment and the desire to clean the invasive species out.

I just looked at the mrhoneysuckle and I can see the benefits of buying that. It's also an easy tool to make with the purchase of a diggIng pole and a simple bracket with some wood.  I don't know if I'd want to drag it all day but I would.

Tordon x 2 (but check regs and don't get in roots of beneficial stuff).  Just hatchet the tree and squirt in the split, larger trees need more cuts like girdling.

We get a bunch of chinese tallow here and try to get rid, but can only control.

I'm surprised the deer don't naturally keep the honeysuckle under control (but you probably have a different species).

Also, remember that areas you can't walk make great bedding areas!

we burn our coastal marshland every couple of years. You can burn green growth, you just have to have an abundance of dead grass from last years freeze underneath to fuel the fire.
As was said earlier, light the fire at dusk so that the dew can help slow the spread.

I would advise you to visit you local NRCS office.  They should be the most techinical people locally to help you with some recommendations on certain invasives and some of the best methods for treatment in the area.  Many of these invasive controls work better in some areas then others.  In addition they will perform site visits and help develop a sound plan for control.  Also depending on the area of the state you are in and how programs work in that area, there will be a good chance that some form of cost share financial help will be available to help you out financially.

Tordon works well if you can get it. Burn baby burn! I have a friend who burns his 1200 acres every year and it looks like a park. The place was heavily timbered 20 years ago and now you can't even tell it.