BLM and NF are different jurisdictions.
Nothing I have read on either NF or BLM sites indicates they take into account the other's turf.
It is workable. Check the rules and regs in the local NF office and the local BLM office to make sure.
If called on it out in the field, don't argue it there.
The regs say "14 days", but it really is 13 nights that you can stay in each place.
The date that you arrive plus 13 is the date that you must leave.
The date you can return to a spot is that arrival date plus 29, the 29th day.
A problem you will have is that you need to find some over night spots to meet the 14 days in a 28 day period rule before returning to the first location. It is the 29th day that you can return.
Example:
First spot:
Arrive 07/02/20 (Return date is this date plus 29, 07/31/20)
Stay 13 nights
Leave 07/15/20
Second spot:
Arrive 07/15/20 (Return date is this date plus 29, 08/13/20)
Stay 13 nights
Leave 07/28/20
But 07/28 is too soon to return to the first spot.
You have to be somewhere else for three nights in each cycle.
An image of this in a spreadsheet is attached.