Report: Raiders haven’t given Derek Carr permission to seek trade

His 2024 and 2025 dead money won't be escalated forward into a single year. If, rather when, he is cut, the Raiders will have dead money on their books for 2023, 2024, and 2025 from Carr's contract. The only way that dead money will total $5.625m is if he is a post-June 1 cut.
No, I think that you have that wrong.

When a player is cut, there are only two options re dead money... (1) pre-June 1st cut requires all dead money (remaining prorated signing bonus amounts) goes against that year's cap, or (2) designating the cut as post-June 1st, which allows the dead money to be split over this year and next year.

There is no case that I know of where there is dead money for cutting a player that extends to year 3 or beyond. It all comes due in the first year (pre-June 1st cut) or is split between year 1 and year 2 (again, for a post June 1st cut, whether designated or not).

In Carr's case, if he is a pre-June 1st cut, his dead money will be $5.625M... and it will all go against the Raiders 2023 cap. After this year, he is totally off the books for the Raiders.