Since you are under age 59½, the only legitimate reason that a financial institution should use code T for a distribution paid to you from a Roth IRA is if the distribution is from an inherited Roth IRA maintained for you as beneficiary. How was this Roth IRA established; was it established by transfer from another retirement account or was it established by a cash deposit of a new contribution?
Assuming that this is not an inherited Roth IRA, entering the code T Form 1099-R into TurboTax and telling 2019 TurboTax that you did not have a Roth IRA before 2015 (even though you did), and also making sure that you don't enter and basis in Roth IRA conversions for years prior to 2015, will cause TurboTax to include the code T distribution on Form 8606 along with the code J distribution and will result in a correct tax return and correct tracking of your Roth IRA basis. You could then move this Roth IRA to another custodian (possibly by indirect rollover to eliminate any possible undesirable coordination between custodians that might cause the incorrect coding to be propagated to the new custodian) to eliminate the uncooperative custodian from the picture. It's possible, but unlikely, that the IRS would question the code T distribution being included on Form 8606. To avoid that small possibility you would have to file a substitute Form 1099-R (Form 4852) and use the correct code J. Filing a substitute Form 1099-R will require mailing your tax return.
