I've never grafted plums, but have done it with apples, pears, cherries. I find the success rate is better with apples than pears.

Last year I made about 20 grafts to a Bradford pear and none took. The year before, as a test I grafted a Shenandoah pear to the same tree with a success ratio of 1/1. I think the problem is that the Bradford pear comes out of dormancy before the scion sellers on the west coast ship their scions. This year I ordered pear scions from Fedco in Maine and I received them in the first week of March, so they're already grafted. The west coast seller I ordered D'Anjou scions from hasn't shipped so I think those scions won't take; if I ever received them. Last year I was grafting in the 2nd and 3rd week of April.

I found that after all the grafting to the pear tree the tree threw out lots of new growth. Lots of long new growth right off the trunk/ So plenty of places for a graft to a stem with the exact same size diameter as my scion. This year I also drill grafted to a seed grown pear tree that produces pears about the diameter of a quarter. I did all the grafts with a drill into the bottom 18" of the trunk, because of vine girdling and deer damage to the trunk. Also hoping I'll get very little water sprouts in the future.

I did my grafts with the following drill bits, I also give the diameter of the bits:

3/16 (0.1875)

#16 (0.1776

11/64 (0.1719)

#18 (0.170)

5/32 (0. 1563)

I think having a few smaller drill bits might make it easier to use the small dia. tips off the scions. Google drill grafting. I got the drill sizes at: this link. Also Heme Depot sells bits for about a third less than in the store and Lowes store prices are higher than Home Depot's. A countersink on the HD page is only $3.57 as I remember.

If I were grafting to your tree I think I'd cut off most of the growth above where they split off the main "trunk". Leave 3 or 4 smaller branches to provide nourishment for the roots. Looks to me like your trunk might be multiple separate trees in a cluster?

I've top worked a few older trees like this, and plum can be grafted over much like apple in the spring. Bark grafts sure if it's slipping, or use other grafts. Since it looks like you have some smaller branches around one inch diameter, I would graft onto several of those and remove the thicker limbs. Remove most of the suckers through the summer and your grafts should put on a few ft of growth, then you can train the new scaffolds.