Let put a numerical example to this so we can all be clear....
Let assume player is given the following simple 3 year contract that includes $3M Signing Bonus, $1M in Year 1, $2M in Year 2 and $3M in year 3
NFL rules allow that contract to be counted against the cap as follows amortizing the bonus over the life of the contract:
-- Year 1 $2M, ... Year 2, $3M, ... Year 3 $4M
The Bills would account for the player in their internal budgeting as
-- Year 1, $4M, ... Year 2 $2M, ... Year 3, $3M the entire bonus counting in the first year
That overstates the first year of the contract and limits other signings whenever signing bonuses are paid. They could achieve the same thing by not paying a signing bonus, making the first year of the contract guaranteed and giving the entire bonus amount as salary.
I don't think the Bills do that. I think they're amortizing the signing bonus contract per the NFL rules, while voluntarily limiting further signings when they've reached their salary + signing bonus cap level (cash to the cap) for a given year, simply as an internal budgeting process.