There's some character to the base of the trunk so if you have the energy and enthusiasm to dig it I'd say have a go. Hard to guess how big it actually is with no size references in the pictures. When I saw the buckthorn trunk so close I thought that's going to be tough but the last photos shows you could dig them both at the same time then, hopefully, separate the roots when they are out.
There are some rules that go with collecting trees though:
1. It will always take more than twice as long as you imagined.
2. The trunk expands as you remove soil so it will be bigger than you expected.
3. The roots will be way bigger than you planned so have a larger container on hand to pot into.
4. You will almost certainly need to dig a much bigger and deeper hole than you hoped in order to cut the roots underneath the trunk.
5. The trunk and soil will weigh lots more than you guessed. You may need help or lifting equipment to get it out of the hole and into the vehicle/wheelbarrow/backpack.
Apart from all that it should go real smooth.
You already know that honeysuckle is real hardy so feel free to chop the straight trunks way down close to the base. Do that AFTER you dig because some long trunks are usually very useful as levers to tilt the trunk from one side to the other to access the roots underneath or to break off any stubborn roots going deep.
Also no problem removing all the field soil from the roots when you dig. That should help with separating the 2 trees.