This question seems to have a number of misapprehensions embedded in it.
I'm not trying to "save" my improvements of the library from GPL
That's good, because LGPL requires (eg LGPLv3 s2) any modified version of the library to be redistributed under LGPL, or GPL.
My project is licensed with a permissive license ... to allow use of the library without strong separation, I'd need the library to be more permissive, at least LGPL.
Well, no, you wouldn't. As the GPL FAQ says
If a library is released under the GPL (not the LGPL), does that mean that any software which uses it has to be under the GPL or a GPL-compatible license?
Yes... The software modules that link with the library may be under various GPL compatible licenses, but the work as a whole must be licensed under the GPL
(my emphasis) The upshot is that you can release your work, in source form, under a more-permissive licence. You must release the binary form under GPL, with the usual obligation to provide sources, but your parts of those sources may be under a more-permissive licence than GPL.
So the only real reason to ask the author to relax the library's licensing terms is if you want to incorporate it into proprietary code, and I would guess said author is unlikely to do that. Nevertheless, you can still ask; it is, as they say, a free internet. However, the conversation is likely to go better if you're asking for reasons that actually hold water.