The only way he has leverage is if the Senators are fighting for a playoff spot, and a player going on LTIR gives them an excess of cap space to use on Pinto beyond what they would get just by sending down a player.
The Senators may lose a draft pick because of the Dadonov fiasco. They are already low on picks if you look at the amount of past and future picks they have traded. So long as whatever Pinto gets on a 1 year deal this year is equal to or less than what the Senators think he will get on his next contract (re: qualifying offer), Pinto could force their hand to get a higher 1 year deal if they have the ability to do it due to LTIR.
For example, if they really think he deserves 1 year 1M, but they have the ability due to a player being on LTIR to pay him 1.5M, and they are fighting tooth and nail for a playoff spot, are they really going to turn down the shot at getting a free rental over a difference of 500k in space, which in this scenario where they don't want to burn a 1st or 2nd on a big rental, they won't use elsewhere anyways?
Without someone being on LTIR, it means they may have to trade someone away to pay Pinto more, which cancels out the idea that Pinto is a rental because they would lose a roster player to acquire him.
The above is moot if they are a basement team or a top team. In either of those situations, he would have a lot less leverage. He needs the Senators to be fighting for a playoff spot, but not comfortable.