The -replace operator uses regular expressions, so you can merge your three script blocks into one like this:
Get-Content $tmpFile1 `
| ForEach-Object { $_ -replace 'replaceMe([1-3]).*', 'replacedString$1' } `
| Out-File $tmpFile2
That will search for the literal text 'replaceMe' followed by a '1', '2', or '3' and replace it with 'replacedString' followed by whichever digit was found (the '$1').
Also, note that -replace works like -match, not -like; that is, it works with regular expressions, not wildcards. When you use 'replaceMe1.*' it doesn't mean "the text 'replaceMe1.' followed by zero or more characters" but rather "the text 'replaceMe1' followed by zero or more occurrences ('*') of any character ('.')". The following demonstrates text that will be replaced even though it wouldn't match with wildcards:
PS> 'replaceMe1_some_extra_text_with_no_period' -replace 'replaceMe1.*', 'replacedString1'
replacedString1
The wildcard pattern 'replaceMe1.*' would be written in regular expressions as 'replaceMe1\..*', which you'll see produces the expected result (no replacement performed):
PS> 'replaceMe1_some_extra_text_with_no_period' -replace 'replaceMe1\..*', 'replacedString1'
replaceMe1_some_extra_text_with_no_period
You can read more about regular expressions in the .NET Framework here.