Congratulations on the upgrade!
I'll be honest....I don't use a dropper for my conditioner, lol. The tiny bottle of Prime with the little dropper is a tad bit easier to use, which was why I gave it to my friend when I gave her a setup for a rescue she took in....which left me with just the normal large bottle of Prime. What I do is pour a little into the cap, then actually....well, dribble it in. xD It takes a tad bit of practice, and I've slipped up a few times and probably way over-dosed a little(I assume I over-dose anyway).....but everyones still alive.
Unless you poured like half the bottle or more in there, its not going to harm anything. I'm not even sure if that much would harm anything.....
So don't worry and don't stress! Everything will be just fine. xD
Now, did you manage to get any sort of caves or plants for Henry to hide in in his new home? Its very important when upgrading to a larger tank especially that there is enough cover to help your betta feel more secure and less stressed.
Also, have you done anything to baffle your filter flow? If the flow of the filter is too strong, it can be rather stressful for your betta since they come from naturally very still waters, plus the added handicap of those long fins that often act like little sails catching the current. While some bettas deal with the flow alright, I've found that most bettas are at the very least not nearly as active with an unbaffled filter as with a baffled one, and a not of them just park themselves in the corner/behind the filter for the most part. I'd suggest rubber-banding a bit of filter sponge or the foot of some panty-hose over he out-flow to baffle the filter.
Also, now that you have a larger tank with a filter and heater....you're going to want to look into Cycing. Here are a couple threads from right here on the forum that should be quite helpful to get you started.
If you do decide not to cycle, there really isn't much point to the filter, and you will need to preform at least one 100% change per week....which can get to be a pain in a 5 gallon. If you cycle though, you can eventually go down to just one partial change a week with a gravel vacuum; much less hassle in my opinion. But up to you.