Yes, to an extent. There is not really any such thing as just “open source” - licenses give rights. With LGPL, you have to give the end user the ability to replace the Processing libs in your plugin with a different or modified version. Obfuscation can interfere with that, so you have to do it with that in mind.
An often used Java obfuscation tool (as you mentioned) is ProGuard. You can probably get it to work with the output of Processing, but you have to ensure you exclude the libs from the process or you’ll be breaking the license terms. Google ProGuard and LGPL and you’ll find more info.
I still find the whole effort pointless, and think you’ll spend more time investigating and trying to get it to work than you’ll get benefit from it.
EDIT - you’ll also probably get in a load of issues with use of reflection across Processing to look up classes, methods and fields by name.