I'm not an expert on licences by any means, but my understanding was that LGPL explicitly allows you to link it to other binaries without needing to licence them with the same licence. Does rust really only support static linking and not dynamic?
Per the Gnu wiki:
Does the LGPL have different requirements for statically vs dynamically linked modules with a covered work? (#LGPLStaticVsDynamic)
For the purpose of complying with the LGPL (any extant version: v2, v2.1 or v3):
(1) If you statically link against an LGPLed library, you must also provide your application in an object (not necessarily source) format, so that a user has the opportunity to modify the library and relink the application.
(2) If you dynamically link against an LGPLed library already present on the user's computer, you need not convey the library's source. On the other hand, if you yourself convey the executable LGPLed library along with your application, whether linked with statically or dynamically, you must also convey the library's sources, in one of the ways for which the LGPL provides.
So as long as you also provide your application with an LGPL library shaped hole you can release a static-linked binary with LGPL components.