On a failed save, worrying about agency is a red herring
Ultimately, the DM decides on what is reasonable, but it would be best to address all of the related bits of your question:
A reasonable suggestion to a creature that failed a saving throw should be followed. The suggestion offered is reasonable: it uses the PC's abilities to do something the player might do to obstruct/harm the PC's enemy. The player has no grounds to object. By the rules of this game, saving throw failures have consequences.
You ask:
In such a case, who gets the say whether the Suggestion is or is not reasonable?
The DM, and certainly not whomever failed the saving throw. You seem to be asking for a hard and fast rule where there isn't a specific rule. The fundamental answer is that "The DM rules on what is reasonable." (But see point 5 at the end). D&D 5e generally relies on rulings over rules. See also "how the game is played:" (PHB, p. 6/ Basic Rules, p. 3)
- The DM describes the environment.
- Player describes what they do {note: dice rolled if necessary ..."the DM decides what happens, often relying on the roll of a die to determine the results of an action."}
- The DM narrates the results of the adventurers' actions.
At a healthy table where the players and the DM generally act in good faith, a player might well ask "how is that reasonable?" and the DM will offer an answer as to why, or will reconsider and then rule "Yeah, not reasonable, so here's what happens."
If the climate at the table is more adversarial then this spell isn't the problem, but rather the problem is the climate at the table.
- Then you ask about adding another roll:
Or should it be determined by a Persuasion roll?
No. The player already failed the saving throw. D&D 5e has tried to streamline combat, not add unnecessary rolls. The necessary roll already happened: a failed saving throw.
- You then add a subjective element to the question
How can the DM give a fair ruling that doesn't take away the player's agency, but at the same time does not allow the player to freely deny the effects of Suggestion?
This is a problem with the question's underlying assumption. Spells like charm, suggestion and command that are cast on PC's -- when the PC misses their save -- by their nature either limit, reduce, or remove player agency for the duration of the spell's effect. That is their express purpose as spells. Similarly, a slow spell reduces their character's movement on a failed save. With that fact-of-the-game's-rules in mind, worrying about a loss of player agency on a failed saving throw from such a spell that influences a PC's decision making is a red herring. The only concern is: is the outcome reasonable?
- What's fair?
Two points on this:
(a) The suggestion that you offered in the example is fair. What the DM needs to ask internally, before choosing the suggested action, is
How does the NPC/enemy know that the cleric had cast Bless?
How does the enemy caster know the cleric has the spell silence available, and would that affect the reasonableness of the spell
What check did the DM make to determine that?
- If the enemy has an idea that such a spell was cast, and that the cleric knows the silence spell, then foiling it in that way is reasonable. If the enemy would have no idea that such a spell was cast, such as the bless already being up before the two parties saw each other, then the DM's problem of giving the NPC omniscience is what's unfair. Likewise with silence. Only the DM can answer that internal question, and a good DM will. (Thanks to @CareySauerbrun)
(b) I refer you back to the climate at the table, and if the tone is non-adversarial or adversarial in nature. The game as written presumes that the DM will endeavor to be fair. What people actually do is another matter.
Bottom line for failed saving throws by PC's
DM makes a ruling, and then you play on. Failed saving throws have consequences.
@Codes with hammer summarizes this nicely, in a comment, as follows:
The PC's saving throw is the PC attempting to determine if the suggestion is reasonable. If the PC fails the roll, the DM decides whether the suggestion really is reasonable.