Balance is Relative
This is less a question of balance and more of how well each character can use it. Your players are one end of the scale and your encounters another. You have full control over the encounters and obstacles on that end, so if you unbalance the player side, you can simply add more to the encounter side. The real problem in balance tends to come about on an individual basis - are each of my players pulling their weight? Is one player way too strong and the rest feel helpless? Or is one helpless and the rest too good? These are issues that can cause player strife. But giving each player the same bump has mostly an equal effect. Can a Wizard make use of advantage as well as a Rogue, and under what circumstances? These are important questions to consider.
So manage your encounters and obstacles accordingly and balance won't be an issue. The difficulty of every campaign is impacted in ways the DMG can't always anticipate, such as how well equipped your party is. So "feeling it out" is an ever present battle, since no book can give you a concrete answer to what is and is not perfectly balanced.
Since the limit on inspiration dice is 1, you should consider what having a free inspiration die means and how your players will react. If you intend to reward them with a die when they do something heroic, they won't feel the need to do something heroic if they can't get rewarded for it. Since they already have 1 die, until they use it they can't get rewarded. You would hope they would do heroic things regardless of the reward, but incentivization is a powerful tool when managing humans in any context. If you remove the incentive during part of your sessions, they won't work towards that endeavor. If you intend to encourage heroism with inspiration die, I would suggest you allow that incentive to be ever-present in your session to a reasonable degree. Consider increasing the limit to 2 or 3 inspiration dice.
With one die, they are incentivized to use it when they have it, but not incentivized to gain more. With unlimited dice, they are incentivized to gain more dice, but not incentivized to use them. Using the original rules of acquiring an inspiration die, the same "problem" arises. Incentivized to use them when they have it, but not incentivized to acquire more. This is less of an issue usually because inspiration dice are harder to come by. But when you start every session with 1 die, you're opening up this problem at the start of every session, which also creates a pattern. This pattern can be useful, but be aware of it.
With a limit of 2 or 3, they can simultaneously be incentivized to use and gain dice to a reasonable extent. It eliminates some of the "overlap", as it were. Discourage hoarding, but encourage the acquisition of dice.