L.S , I've had a chat with PM accordingly, and want to share the good news immediately:
We've decided based on your Feedback and internal discussions to indeed implement a way to easily replace the name "Enscape" in batch exported render file names. This feature will present options to instead select the project name, company name, or simply a name of your choice instead. We're of course still fine-tuning the plans behind this function so there may be even more options available as well.
Overall this is something we plan to add with 3.4, or if time and development resources allow for it perhaps even 3.3 (Or a preview prior). No specific promises I can make yet besides that this will be surely added! Thank you all very much for the input again.
Thank you for the prompt response. Please also give an option not to include time stamp in the file name. The whole point is to rewrite the existing render files so that file-linking with other programs remain intact.