my logic will differ than those of SouthFayetteFan, Lain and of striker1064
Amex will give u cards like candy at the beginning but judging by recent DPs in here, someone in here had only 3 Amex cards and was denied bonus on her 4th app.  I think many ppl in this forum had similar problems but i dont recall who.
When it comes to Amex, it was show via a reddit survey that people who tend to churn amex, 40% of ppl would only spend between $100-$600.  I believe this is a huge no no with amex.
now another proof that ive seen is a friend of mine who MSes very hard.  he puts close to $100k a month, sometimes peaking at $150k.  he basically maxes out every single amex cards and then goes onto his other cards... He recently got a Amex Biz card, although he was denied with a 680 score, he still ended up approved later on.
I also havent gotten the message so far and ive been hitting amex cards hard.  started 2 years ago with a SPG Personal card, currently have 5 Amex cards applied since then.  i have spent close to 35k in 6 month last year on my SPG card, was given the 100k upgrade too.  I think end of the year, i had close to $60k total used on amex card, i have yet to get the pop up.  *knock on wood*
As for your first choice, would recommend getting a amex plat via cardmatch which would net u 100k, get the BBP next, since it is a nice non category card, hit that card hard, then slowly start churning the rest... stop if u get the pop up and just try to put as much spendings into Amex for the popup to go away.
Thats just my 2 cents.