Sorry to say, you are out of luck. You cannot PC a card that is under one year and with the BC you can only upgrade to BCP and no other Amex card.
You would need to reapply for another card and at the looks of it $500 CL is just getting barely approved.
Personal Cards: Amex Plat | Amex Delta Res | CSR | Citi AA Exec Business Cards: Ink+ | Amex BGR
Congrats on the BCE! 
@longtimelurker wrote:Why is the BCP better for you? The major value over the BCE is the much higher earn on supermarkets (and a little on gas) and now some streaming. If you don't use supermarkets much, is the $95 AF worth it on streaming
$132/month on streaming and you're already over the AF! 
Who needs a cable bill when you can have all the streaming services! ...I kinda wonder how that'd work out, honestly. Maybe not $132/month worth, but loading up on services...you could get pretty much everything on demand and ad free. Many services get shows not long after they air on tv. And some even offer live watching.
I assume OP is 21 and living in a city. So the BCP having a transit category may be enticing. Daily transit really adds up. Toss in 2-4 streaming services and occasional groceries (can always just snag gift cards, too) and the bcp could work without eating in too often. Granted better options are out there.
Back on topic, though...!
OP, don't fret! The important thing is that you got your foot in the door with amex! Now you can build up with them...all with soft pulls! Just keep the card open. Let it age and help you pad some age to your credit report & payment hsitory. When you have an AMEX card, they'll just about only ever do soft pulls on you when applying for another card or requesting a credit limit increase.
Keep the card. In 61 days, do a CLI request to 1500. 6 months after, 3x request again for it to be 4500. Now you're really getting somewhere! Then in 4 months, your card will be a year old and be eligible to change into the BCP. Or maybe apply for the Gold and Green (the green card is awaiting a refresh...and the rumors about what it'll offer seem to be right up your alley) cards. Just keep the bce in a drawer after getting the Gold & green, or do a product change to the cash magnet, which is $0 AF and will give you 1.5% cash back on your odds and ends purchases. And just 2 more months after your product change...your card will be ready for another cli to be well over 10k! 
In 14 months, you could be opening a total of 3 cards, 3 CLIs, and a product change...all off of that first original hard pull--not bad, if you ask me!
I did this with my PenFed card. I applied for the Gold when I meant to apply for the PowerCash Rewards. I called PenFed and the guy said I would have to apply for the other card, but that it would only be a soft pull and I could combine the two cards once I received them and close the Gold.
You can kinda do the same with AMEX. I have both the BCE and the BCP (although I will be closing the BCE soon). Since you already have an AMEX card, any other app will be a soft pull. You can check the BCP and see if it says, "You're Pre-Approved", which means you're pre-approved, or if it says, "Expanding Your Membership with Us is Easy" then that means there's no pre-approval. You can app for it anyways because it'll only be a soft pull, but it's not a guarantee you'll get it. After a few months, you may be able to close the BCE and move the credit limit over, although the limit on your BCE will have to be more than $500. Mine started off at $500 and I got an auto increase a few months later. Check out AMEX CLI threads to find out how to go about getting one with your AMEX card.
Long story short, you'll have to app for the BCP, wait for a CLI on the BCE and then move the limit over, or just wait a few months to see if you can upgrade it.
