Nope. You as satellite operator are responsible for requisite communications permits as a pre-requisite for your launch permits, including Earth to Space, Space to Earth, and Space to Space, all of which are separate item and while you can put any combination of multiple ones of them in a single application to the regulatory body, you end up with separate permits for each. In that scenario You would have to get a bunch of information from spaceX in order to prove to the regulator that your not going to interfere with the operation of the entire starlink constellation (spaceX providing the hardware is not sufficient proof). What SpaceX could do is provide a module that connected to the StarLink inter-satellite laser links they are planning to add in between the StarLink satellites.
Like I said, there’s really only two scenarios to change this. One is that we get regulatory change, which the ITU (the people globally in charge of radio regulation between countries, the FCC and ACMA coordinates their respective countries compliance with ITU regulations in order to meet extremely well established treaty obligations, the ITU is over a hundred years old) rejected the need for in their last meeting, based on evidence gathered in the years before that, essentially this entire scenario will break down when Starship slashes launch costs and they will either rule with an iron fist or bend to allow ISM band style approaches in space. The other is if
laser communication systems come down significantly in price, the current cheapest I’ve seen is a one way (space to ground) link for “$10000 USD” list price and that wasn’t available ready to order and had no flight heritage so I’d be unsurprised if the price became $25k USD when it finally had some flight heritage. (One of the nasty secrets of aerospace engineering is how much Commercial “Off The Shelf” hardware is actually just built to order based on an approved design) and you need two way units to actually become cheap enough they aren’t more expensive than a radio permit is, which they aren’t at the moment since most people would have to both get the satellite hardware in the order of a hundred thousand or more, and also build a dedicated observatory with a sufficiently good telescope and powerful enough laser as their ground station which will probably set you back at least $50k unless you score as bargain or strike a deal with someone operating an experimental one.