[Image description: a four panel meme of fictional character Poison Ivy dancing, science-ing, and smiling with a beaker in hand. In the middle of the image is text that reads "POV: When I successfully propagate those expensive cuttings"]
Don't give up, it'll happen! I've only grown a few honeysuckle varieties but they're fairly amenable to layering, if you've got the mother plant available to work with. If there's a honeysuckle species in particular, let me know and I'll check my books for you (it's Dirr's Reference Manual of Woody Plant Propagation usually)
Not sure on the variety, but it's wildly growing here in southern NJ near Philly - I foolishly thought I could just clip it and use some rooting hormone, but it seems like it needs a little more TLC than that (such as making sure to clip only new growth?)
You got it right, softwoods are the better choice for rooting with hormone, be sure to keep it misted as well. If it's got larger leaves, don't be afraid to cut them perpendicular to the main vein to reduce water loss.