Demo92,
I deal with controlling honeysuckle as part of my profession. Just a few things "Bush honeysuckle" is a native to the northern part of the US. What I think you're refering to is Exotic Honeysuckle species, sometimes refered to as Bush Honeysuckles. The best and cheapest way of controlling larger shrubs, anything 8 feet or more, is to cut them off with a chainsaw about several inches from the ground and treat the stumps with a 50/50 mixture of 41% Roundup and water. Make sure that you treat all of the stems. By adding die to the mixture, you can see what you've treated easier. This can be done anytime of the year, except during the spring growth period when the sap is flowing upward. For shrubs 8 foot or less, you can spray the foliage with a 6 oz per gallon mixture of 41% Roundup mixed with water. Just use a hand held spayer and spray a fine mist on at least 80% of the foliage. The best time to do this is during the fall after most other native species have dropped their leaves and honeysuckle is still very green. In order for this or any other control program to be effective, consecutive years of treatment is needed. I would monitor the infestation yearly untill it is under control of gone. Hope this info helps, I've had amazing success using the above mentioned methods.