I think the easiest way to balance things out would be to have a rule that if the player is on LTIR on the last game of the season they are ineligible for the playoffs. Make sure they're a cap compliant team heading into the playoffs. Really discourage the players (Kucherov) that might be fine the last week of the season but only come off on Day 1 of the playoffs. And to discourage teams trying to stack the deck for the playoffs.
Actually this isn't the worst idea but I'd tweak it.
A team must have a cap compliant roster at the start of the playoffs based on each player's respective cap hit to that team for the season and any player on LTIR is ineligible for the playoffs.
Teams have a reasonable idea on if a player may be available for the playoffs. Maybe it's round 2 or 3, but if the player is important enough to the team, the team will make room. Yes a team would be shorthanded early on because of injury, but those are the breaks. They'll have to weigh the importance of the player vs. short-term gain in earlier rounds. Team's during the regular season also have to maintain unused cap for a player scheduled to return from injury, this isn't that different.
Still removes the cap for the playoffs. There's too many reasons why enforcing a cap through the playoffs is just too complicated and unlikely to get buy in from all parties involved.
This would accommodate for all the various reasons why enforcing a cap during the playoffs won't be easy. Takes into account teams accruing cap during the year as each player's cap hit will just be what their team has taken on. Still counts buyouts and retention in the calculation. Still allows for teams to make call ups and have flexibility during the playoffs in case it's injury.
Doesn't force a player back like with the suggestions a player must play in game 82. If the player is injured, then he's injured. No point forcing him back, but it does mean a team will be shorthanded, that's just how sports work.
LTIR is meant to provide relief, not complete replacement and to give the possibility of going over cap during playoffs. Pretty sure that's just an unintended consequence.
Unless there's good reason why this idea doesn't work or make sense, this is probably my go-to idea to fix the issue. Also way more likely to get agreement from the league then my way more fun previous suggestion to remove the cap after deadline...