If the GPLed files are included in the build, or linked in, then the whole thing is GPL, not LGPL. You can't change the license, and you can't use GPLed files in a build without releasing the resulting binary and source under the GPL. (You can license your own source files under more permissive licenses, if you want.)
Therefore, the whole distribution would be under the GPL unless you either replace the GPLed portions or get permission from the copyright owners to distribute under the LGPL, which is worth asking about.
You don't need to provide the source used to compile in the distribution, but if you don't you need to satisfy one of the other ways. One is to make sure the source is available along with the binary, and one is to provide a written offer to provide the source for a nominal fee for at least three years after you distribute. Your choice; you don't need to do more than one of these. However, all the source must be in the source distribution.
I have no idea what you mean by a "compile with GPL code" option.