Forget about a line trimmer. The rotating plastic string thing? There are brush cutters with rotating metal disc cutters but I've only used those like one would use a scythe; a fast swinging hack. Doesn't sound applicable.
There are 'baby chainsaws' used to prune trees way up high. The ones I've used are on long poles that would make what you want to do unwieldly and tiring.
My first thought was using a hedge trimmer; 2 reciprocating long thin blades. Easier to put between the ivy and the fence or tree and pull away from the fence or tree. Then you said there were branches the size of your forearm. You the hedge trimmer works fine for up to 1/2 inch diameter material but not the size of your forearm. Depending on how many of these thick mature vine stems you have to cut (a pruning saw will give you a workout) possibly the use of a small chainsaw (the 'baby' size) combined with a hedge trimmer. Both of these can be found at a good rental company.
I'd use the chain saw to cut the ivy vines as near to the ground as possible. If you allow the separated vine material to die, it will be far easier to pull off the tree and fence later.
I had to go peek at what is newly available these days. This caught my attention and it is only $80 bucks! And although gas equipment is my go to choice, you pay the price for better power in weight. For what you are doing this looks like...the 'bees knees'? The right tool for the job? You want one tool that you can control easily. Take a look and let us know what you think. Do you want to kill all of it? Leave a bit on your fence? Killing the roots is another part of your problem but we need to know more details such as how close other plants to your ivy and roots? Please send a picture or two.