a systemic will damage the parent plant. so there isn't an easy solution.
The best way is perhaps to ask the neighbour to 'tear' the suckers off the root stock on their side.

I was hoping that lilac was vigorous enough that it would hardly notice the glyphosate in the parent plant but the suckers would wither...
Unfortunately the hedge is preventing easy access to suckers from either side, there is a 12" wide pathway for trimming on our side (between the hedge and a shrub border) which gives the best access, and I'd seen at the GC a long-reach weed spray thingummy I was considering buying.

If all you're going to do is spray the sucker growth with glyphosate rather than cut into them and apply SBK or similar, I doubt very much it'll kill the tree. In fact, I doubt very much it'll actually kill the suckers, but I'd do it anyway. Glyphosate sprayed on woody growth doesn't kill it, it kills through the green, and the small amount you'll be applying is unlikely to end up back in the main tree.

That's the answer I was hoping for, but I figured it was maybe wishful thinking.
Have you ever tried it on lilac suckers?

No - I tried it on a self set elder about 9 inches to a foot high. I couldn't dig it out because the root had gone under the shed. Well I sprayed and sprayed and sprayed - the leaves keeled over but always regrew. Wasn't round up though, just basic glyphosate. Gave up and poisoned the blasted thing with SBK in the end...

Hmm, sounds like I need to learn to love the lilac then.
Thanks for the advice.