How to rename files by moving/shifting characters in file name (e.g. appended sequence to prepended sequence)?
I'm not familiar with either of the programs you've tried, but here's how to do it in A Better Finder Rename ($20), and I expect at least one of yours can do it in a very similar manner. See below the screenshot for instructions and an explanation.



First, I chose ABFR's "Re-arrange using regular expressions" feature. This will probably be named something similar in other programs. This requires a regular expression to match with, followed by a substitution pattern to create the new filename.

(.*)_(\d{3})\.mp4



The first capture group (.*) simply matches any number * of any character . necessary to make the filename match.
This is followed by a literal underscore, which in your case delimits the border between the filename and number.
Then, the second capture group (\d{3}) matches exactly three {3} digits \d. If some of the files have more or fewer than three digits, say a number between n and x, you would change it to (\d{n,x}).
Finally, the extension is matched with an escaped period \. and mp4. If your files have different extensions, you should change this to \.(\w{3}) — just like the number, but with characters \w instead of digits.


Then, we use those capture groups to make the new filename. The syntax for this part may vary more between programs.

$2_$1.mp4


This is perhaps more straightforward: our second set of parentheses surrounded the number, so we put that first with $2, followed by an underscore, the filename $1, and the extension .mp4. (If you used \.(\w{3}) earlier, write $2_$1.$3 instead.)

If you'd like practice and help with writing regular expressions, I recommend the website RegExr, which provides live syntax highlighting and tooltips.