Unfortunately, many landowners are going to be in your boat. Spending thousands to combat a plant that some folks still promote. Even on the properties where people claim honeysuckle is the best species for deer because of some other circumstance, such as those laid out above, you have to remember that while that may be the best option for deer (debatable) it certainly isn't for any species that requires succulent plants or sunlight.
I don't have much issue with any non-native plants that aren't invasive, your property, your choice. However, invasive non-natives are problem for everyone. I battle reed canary grass over large portions of my property every year because someone upstream thought that was a good idea to plant.
I also realized I never answered OP's question regarding the best way to trim BH, I would recommend cutting it to three inches off ground and lightly spraying stump with Triclopyr/Imazapyr mix (I jest).