You do have to be careful with this one. On the one hand, it's a good way to encourage people to play the game despite not having many cards. On the other, the players that visit that FLGS will probably still turn up, due to GNK prizes or wanting to try the 'unique' meta or whatever other reason, and it could well be that if they don't turn up, you'll find it hard to get enough players. It mightn't be super good (or the reason for the tournament idea) for the beginners to be matched against the experienced players, even with the limited card pool.
I think I'd like to note that GNK prizes for a beginner's event do seem a bit out of place. I know that @tomdidiot is running a beginner friendly tournament split into two pods (one full card pool, one restricted to one core + one deluxe). I'm not sure how the GNK prizes are being split, but I think the top prize from the restricted pod is Opening Moves (though could easily be "data pack of your choice"). I like this format a lot, it lets experienced players play full Netrunner, and beginners play Netrunner for prizes that they actually want! This also stops problems I brushed over in the previous paragraph: experienced players won't be paired against beginners and probably won't want to enter the limited card pool event. The only limiting factor here is number of players, it'd really suck if you only got two or three beginners turn up with 20 people coming for normal constructed.
As an addendum to this, you don't want to have too many beginner-only or restricted card pool only events. Your players will want to play some real GNKs, too. I don't know how common the beginner events should be and I love that they exist, but you certainly should be careful: hopefully without sounding too entitled or whatever, I don't want all of the GNKs I go to to be Core Set Only. FLGSs only have so many times they can run Netrunner tournaments, and FFG only send a certain number of GNKs per season.
Something my FLGS does is have an all-day casual play session every month we don't have a tournament, which is something I'm not sure a lot of other FLGSs do. They're free, no-registration, turn-up-and-play, and they incidentally tend to be great environments for beginners, whilst being fun for more competitive players too (without intimidating words like "tournament"). We get a lot of casual players there that we don't get at the tournaments or weekly events at the pub!
I like this idea a lot, but something I think would be even better (perhaps as well as?) would be encouraging the FLGS to display something along the same lines to that, or the ANRPC Code (in short: "don't be a dickhead") on the wall. Perhaps, if nothing else, you could put it next to the standings for only the length of the event?