The bad smell is from oleic acid. @JeffH is right, bees find it repellent. I would remove them too
I’ve done this with success in the past so I encourage you to give it a try! If it’s not too cold you could place a frame of open brood between frames of mostly drawn comb so it’s better insulated from temp & movement, in a lightweight box that is closed on the bottom and with a partly open top (or all the way open/off altogether if warm enough) and hoist it to just under or next to the swarm. If you can manage that the bees will move in in no time.
Just an update! The swarm stayed in the tree for over a week and endured 10mm of rain one night. I had several swarm members visiting the bait hive they originally went into on a daily basis. I think they were getting a little respite from the weather. I gave them a few drops of sugar syrup on top of the frames to keep them going. More and more turned up for the free feed and I hoped they might migrate down. I also added more hives about the place and put them higher but there was only passing interest. A few times I saw a number of scouts about. I think the issue was height rather than dead bees (the dead bees were months old - pulled out of cobwebs that cover every inch of the place pretty much). I totally agree - recently dead bees reek. I will, however, leave it out of my unsuccessful formula.
Sadly, the swarm left on Thursday when the weather was finally favourable. It was about 1/3 -1/4 the size of the original. I think the birds and at least one blue tongue (AKA shingle-back) feasted on the bees dropping out of the cluster onto the ground below. I haven’t seen a swarm move like this one. They were all over the shop and spread too far and wide so I suspect they didn’t all make it to wherever they intended