Getting weight measurements INTO Garmin Connect - 3 ways only?

I'm using Wight Logger android app - need manual sync but works like a charm.

That's not exactly true.  First of all even if you transferred body fat and muscle information it wouldn't show up because I don't think the Garmin database is set-up to use that data.  Second, Garmin only appears to accept one entry a day, so if you weigh yourself multiple times only the last entry for that day will stick.  

Now as far as autosync, my initial experience was not it doesn't autosync, but then a few days later I started noticing that it is syncing the most recent weight.  I'm not sure where it's getting it from, but I did set up sharing with runkeeper, myfitness pal, and Strava and I think all of them pull my weight data from Withings/Nokia. 

So yes autosync appears possible and having a more extensive script probably doesn't add any value unless I'm missing something obvious.  

Sadly I cannot tell you which app permitted to autosync, but if I had to guess it would be myfitnesspal, even though the data sharing clearly DOES NOT say it share's weight data.  Go figure.  

have you tried submitting your concerns to https://www.garmin.com/en-US/forms/ideas/
I had a chat few times on various issues and twice I was told that Garmin actually consider doing something when they get a lot of feedback on the same issues and they advise me to submit my ideas using that form which I did. So I guess everyone who'd like better integration of Garmin with other fit apps should drop them a note and ask.