Button to print embedded report without showing the report on the form

Yes you can do with in one Formula URL button, but it's a two step process to make the button.

You can make a URL formula button to run any report.
A report link field is just a built in feature to run a report with one extra filter - that being how the Report Link field is configured, typically but not necessarily where the Record ID# field on the parent matches the field [Related Parent] on the child record.

So, first step is to make your perfect report with an <ask the user>  for [Related Parent]
Run the report, answer the question and observe the URL.

Now make a formula URL to mimic that URL

URLRoot() & "db/" & [_DBID_of the child table found in advanced properties] & "?a=q&qid=99&nv=1&v0=" & [Record ID#]

So translated that says:
Go to my Realm and then to the child table and run report #99 which has 1 question and the answer to the zeroth question (don't ask me why but computers sometimes like to start to count from zero) is [Record ID#] of the parent record.

OK so first get that button working to run a report of the child records.

Then once that is working add this to the end of the code

&dlta=pr

That will cause the report to print.