No when you decline his option you lose his Bird rights.
This is not correct. Bird rights are based on how long a player has been with a team without changing teams in free agency. They actually carry across trades, the only thing that resets Bird rights is signing a free agent contract with a team other than the team you were on most recently. Bird rights accrue in three years (Early Bird in two) and allow the team to exceed the salary cap to sign the player as a free agent. With full Bird rights, the team is allowed to sign a player to a 5-year deal at that player's maximum salary, regardless of the team's cap situation.
Giles has not (yet) changed teams in free agency, and so whatever team he is on at the end of the year (us, or a team we trade him to) will have his full Bird rights.
HOWEVER, because his fourth-year option was declined, we (or any team Giles is traded to this year) are not allowed to sign him for next year for more than the value of the option year (which was about $4M) regardless of the fact that we hold his Bird rights. If Giles does sign a one-year, $4M contract with us this offseason, then we would be able to offer him any contract up to his max in the summer of 2021 without regard to our salary cap.
Very small chance Giles is back next year.
This is more on point. Giles' max contract this offseason will be somewhere in the high $20Ms. But obviously nobody will offer him that. I don't know how much cap space teams are looking to have next year, but most likely the best offer he can hope for is in the range of the MLE, which will be about $10M or so. Assuming that we want him back, Giles will be back if
1) Nobody offers him more than $4M
2) He gets offered more than $4M but prefers to stay (perhaps to keep his established Bird rights which would allow a much bigger contract from the Kings in the summer of 2021)
3) Something weird is going on under the table
The Giles situation this year has been, frankly, bizarre, and I can't begin to understand it. It seems there is a lot that went on behind closed doors and we don't know what that is, and whether it is bad for our chances to re-sign Giles, or perhaps not so bad for those chances. What we do know is that we can't offer him as much money as the other 29 teams in the league, if they want to. The rest is obscure.