No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
I think you have an extremely good chance. Be sure to find the best signup bonus you can for the Platinum, may need to use incognito/private browsing to do so. Also, kudos on really thinking this through - you and your gf seem perfect candidates for the Plat and will certainly reap the benefits of membership.
From recent experience, it's doubtful you would even get a HP for the Platinum (but it can happen). Amex has usually not been doing HPs on existing cardholders at all, and previously only if you were approved. (I got my SPG in April and my last HP from Amex was in 2002 or 2003.) Also, it's an open secret that all of the personal charge cards in the green/gold/platinum family have the same underwriting standards, and those tend to be lower than the revolving cards. If you can qualify for green you can qualify for platinum.
The $500 SL was certainly a marginal appoval, but there's no harm done in asking for a $1500 CL. If denied just set it on your calendar to ask again 91 days later. If approved, set it on your calendar to ask for $4500 181 days later. To be clear, I am not too familiar with the credit steps from Amex (that's a fairly new development for them) so someone please chime in if the usual 3x guidelines don't apply here. Regardless, I don't think asking will have any bearing on your charge card application.