As others have already replied no iTunes does not itself have a built-in LAME encoder.
I do seem to recall in the past managing to install a codec which as I dimly recall allowed LAME to show up in iTunes, I think it was a QuickTime codec. This approach is I believe long broken, Apple no longer really support additional QuickTime codecs.
However I did find this https://code.google.com/archive/p/blacktree-itunes-lame/wikis/iTunesLAME.wiki I have no idea if it is still compatible with the current version of iTunes and OS X.
Actually thinking more about this perhaps it was this second approach I used to use. I now keep all my music in Apple Lossless.
It looks like it takes a slightly different approach and via an AppleScript gets iTunes to use an external LAME encoder.
It should be noted that an MP3 file produced using LAME and an MP3 file produced using iTunes are both MP3 files that can be played using any MP3 player - otherwise they would not be MP3 files! The only difference is that allegedly MP3 files produced using LAME are superior audio quality to MP3 files produced using other MP3 encoders including the one built-in to iTunes.
Otherwise the approach suggested by others of using a different program to take the recording from iTunes and converting it externally to iTunes is the best approach. XLD does indeed support converting to LAME - it is slightly counter-intuitive that it does since XLD is predominantly for handling lossless formats rather than lossey formats such as MP3.
You don't detail how your recording the audio book. Ideally you want to record in a lossless format or record directly to LAME. You want to avoid any extra conversions, for example recording to MP3 or AAC and then converting to LAME would actually impair the quality. However recording to a lossless format WAV, AIFF, Apple Lossless, FLAC and then converting to LAME would be fine as you are not converting from an already degraded format.
It looks like it might be possible to use Audacity on a Mac to record directly to LAME. However the LAME module is an extra optional download.