You can grab the HH Amex at no risk (i.e., no annual fee) and use it to pay for your HHV stay. I would get that card now if for no other reason than to use it for the stay and collect the bonus HH points.

You can probably also get the AA Mastercard with 5,000 or 10,000 bonus AA miles - more than enough to compensate for the annual fee of $50. Whether or not you choose to keep it for more than a year depends on your CC spending habits and your long-term interest in AA miles. If you can really drive a lot of living expenses to an MC/Visa, the AA card is probably a better option than an HH Visa. For most people, 1 FF mile is worth more than 1 HH point. Again, consider the annual fee and whether you are a good candidate for a "home-grown points" card before you decide to carry the AA card long-term (beyond 1 annual fee).