Amex has a 4 or 5 credit card limit. Business cards do count. The only cards that don't count are charge cards (such as Amex Gold and Amex Platinum), where you have to pay the full amount each month (unless you separately enroll in Pay Over Time). So in case they deny you for going over the 4 card limit, tell them you've heard it's a 5 card limit now and ask them to reconsider. But if they refuse, you either have to negotiate to close one of your cards to all this own to open, or give up. But don't close any Amex cards ahead of time, that may cost you your signup bonus:

https://www.flyertalk.com/forum/amer...ne-2018-a.html

Amex USA terms now penalize:abuse,misuse,gaming;1st yr downgrade/close;leaked offers.

And make sure you've done spend (and had it post to your statement already) on each of your \4 Amex cards recently enough. Amex is denying people signup bonuses if they find that you've been "sock drawering" your existing Amex cards!

So if it's going to take time to put your Amex house in order based on the above, you may want to try at least one of the Chase cards first. Keep in mind that you'll have to do minimum spend on each card, so you may want to space those out anyway. Amex, in particular, may also deny you a bonus if they "sense" that you bought gift cards to meet the minimum spend requirement; I don't know about Chase (I do only "natural" spend and/or at most occasional gift cards bought at grocery stores with some groceries mixed in on the same purchase, so that it doesn't "small" like a gift card purchase based on the total charge amount; ie, you don't want $505.95 ringing bells loudly on your statement.). Definitely don't buy gift cards form a website known to mostly sell gift cards, that's automatic denial pretty much with Amex these days.