Yes, it's closed source. I can't see a realistic alternative to this given I'm trying to make a living selling the app.
But FWIW, I can promise you I don't transfer your browsing history. My policy on such data is I simply don't want to ever have a copy of it. If that means I can't sync your history between Link Bubble installs, so be it.
I think an interesting middle ground would be to connect to existing browser sync storage. Personally, one of the (few) disadvantages of Link Bubble is that my Link Bubble history is separate from my main browser history (I don't really want to remember whether I read something in Link Bubble or elsewhere if I'm trying to find it later). Connecting to existing sync services would be a nice way of dealing with that.
But FWIW, I can promise you I don't transfer your browsing history. My policy on such data is I simply don't want to ever have a copy of it. If that means I can't sync your history between Link Bubble installs, so be it.