You don't have to be pre-approved, just a bit of extra security.. Honestly if you want a BofA card I personally think you would be fine, but the pre-approvals look like this I have a few now, but nothing of interest and the APR's are on the high end for me although i try to PIF monthly.
That is what a BofA pre-approval looks like although 99% of my cards I have gotten have never had pre-approvals other than Cap1 and a few Chase cards. So don't put to much weight on those..
Since you have an Amex card you are probably pre-approved for any of their card and the worse case on new cards they are only SP for approvals or denials, it really comes down to whether you can use another Amex card and what value you want from one of their cards if they are appealing to you or not. I usually myself go for cards for their sign-up bonus and then use most of my spending on a 2% card there after such as a Citi Double Cash card to maximize my spend or if eating out on the CSR, etc.. Really depends on your need as we can tell you what you want in a card with regards to rewards, etc... I think you probably if you want a BofA card would be fine personally even if not pre-approved, fine with Amex as well or probably most other lenders like Citi (although more conservative on CL's), etc... Best of luck on whatever you decide to do and hopefully some of the info was helpful