The thing that you are saying about the integration peaks disappearing unless you choose "MS level 1" in quantification settings does not sound right.

Can you send us your Skyline document?
In Skyline you can use the menu item:
File > Share
to create a .zip file containing your Skyline document and supporting files including extracted chromatograms.

The Share Document dialog gives you the option of including the raw mass spec data files in the .zip file, which might be helpful in this case.
Files which are less than 50MB can be attached to these support requests.
You can always upload larger files here:
https://skyline.ms/files.url

After I see your Skyline document I might have a better idea of what is going on.
-- Nick

I see that you have already uploaded your Skyline document.
Perfect!

I will try to figure out why Skyline deletes your MS2 transition when you make any change to the MS Level in the Quantification settings. These two things would not be expected to have anything to do with each other.
-- Nick

Thank you for uploading your Skyline document.
I see that on the "Instrument" tab at "Settings > Transition Settings", the "Min m/z" value is 100.
Because of this, Skyline will not allow you to have any Transitions with an m/z below that.
Normally, Skyline would not have allowed you to ever add a transition like this in the first place. Now that you are in this state, any change that you make to the settings would probably cause Skyline to remove all transitions whose m/z is less than that Min m/z value.

I think if you change that Min m/z value to a smaller number like 50, then you will be able to change the "MS Level" in the Quantification settings to either "All" or "2" without your transitions disappearing.
-- Nick

This was it. Thanks so much!