I can't really help you. You need to see a good qualified accountant.
First, your issue appears to be with the payment processors. You could apply for a tax exempt letter under 501(c)3 if you wanted to. Because you choose not to apply by invoking 508(c)(1)(A), I can't tell you what you would need to do under the terms of service of the payment processor to not be issued a 1099-K. You may want to review the services offered by the IRS EO office.
https://www.irs.gov/charities-non-profits/exempt-organizations-reporting-changes-to-irs
https://www.irs.gov/pub/irs-pdf/p1828.pdf
You likely should apply for an EIN, otherwise you will have to use your personal SSN to receive your payments. You can do this online at the IRS web site. (Note that the person who applies for the EIN will be listed in the IRS records as the responsible person, and if there are any questions about the church's taxes, they will contact that person. Once you have an EIN, if you later need to change the business name, address or the responsible person, you can send a letter to the EO office informing them of the change.
I have some concerns with certain services and benefits being contingent on a specified fee. I realize that some churches today do this routinely and seem to get away with it (e.g. Scientology), and I know that in the past, church members were often expected to make specific donations, and sometimes families even bought their own pews (as in a church pastored by my great-great-grandfather around the turn of the 20th century). But I can see why the payment processors might be uncomfortable with the arrangement.
If the 1099-Ks are issued in your personal SSN, there is a way to deal with that on your personal tax return, but you may get a letter from the IRS. If you obtain a church EIN and 1099-Ks are issued, the church could file a form 990, although it is not required of churches. If you get 1099-K forms and don't file form 990, and if the IRS then sends followup letters to the responsible person for the EIN, then you would have to write back and explain. Here, I think you would be best served by an accountant who specializes in church tax matters.
