From my point of view there are two misconceptions in your post. 1) you need to be set up to work-for-hire to write an invoice. 2) you need to get clearance from their employer for things outside work hours.

ad 1) No, you don't need to. At least in Germany anybody who's legally competent can write invoices. If the invoices are secondary income, you will be taxed heavily (and declare it you must), but that's it. It has been some time since I last lived and worked in the USA, but I mean to recall that it was basically the same. Of course, invoiced money is your money now and you need to donate it to the FOSS project, which then needs some kind of treasury. But you said as much already.

ad 2) No, you don't need to. Your employer is your employer, not your owner. Now I don't know about the USA today (see above) but in European countries what you do outside working hours is your private affair -- discounting a few, very specific fringe cases. If you play soccer, dabble in explosives, or code for money doesn't matter. And frankly, your typical employer in most cases does not care anyway.