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Coinbase Commerce needs no introduction but I think they don't have custodial wallet like you want.
Although we're talking about Coinbase, Coinbase Commerce is non-custodial service and OP explicitly mention it's not what he need.
P.S. i don't know it works as non-custodial service, but i guess they require you to give the master public key.
Coinbase does not work because, if I am not mistaken, their API is only used to receive payments, it is not used to send payments.
Their service is non-custodial, so it's impossible to create API to make payment. At most, the API will make unsigned transaction where you need to sign it (with wallet software such as Elecrum).