Yes, you can manually tag the putts with the iPhone app, the Apple Watch app, or on their Link belt device. (I don’t have the link—I use either the watch or iPhone.) In fact, it’s pretty common for me to adjust the # of putts as I finish a hole, since I often have to add on the tap in putts—I don’t have the patience to let it register a tap in.
You do have to pair the putter sensor with the app. If you don’t, the system won’t work at all (just the way they designed the system, I guess). Found this out the hard way: my putter sensor died, and I tried to re-pair it, unsuccessfully. No big deal, I thought. Except next time I went to play (replacement not yet arrived), it wouldn’t let me start the round: “need to pair your putter.”
Next time the putter sensor died, I knew better: ordered replacement, but didn’t un-pair the dead sensor. Arccos thought I was chipping in every hole, no putts recorded. I just added the putts manually as I went.
So for you, pair the putter sensor at home (a onetime thing), put it away in your desk or wherever—don’t throw it away, because you’ll probably need it sometime in the future. Then add all putts manually, as I did when my sensor died.
Here are what the iPhone app and Apple Watch app screens look like (don’t know what Android looks like—I get the impression it’s set up better for Apple):
[Yes, I’m on my third sensor now. First one lasted about 5 months IIRC. Second one only lasted 23 rounds. Both replaced free under warranty — forget if it’s 1 or 2 years. But I’m a little concerned they both died so quickly.]