Again, if you are not using port 25, you don’t need an MTA. Period.
MTAs are for sending mail from one mail server to another. That’s not what you are doing. You don’t need to be running your own mail server. You are sending mail to an MSA (Mail Submission Agent). Mailgun is the MSA you are submitting to and the MTA that will handle transferring your message to your destination MDA (Mail Delivery Agent).
If you were submitting mail to your destination server over port 25, you would need an MTA.
But, as you stated, you are submitting mail to an MSA using port 465 or 587. You need an MUA.
This terminology is important, because it determines what role the software you’re using plays and how you need to configure it.
There are plenty of tools that act as both MTAs and/or MUAs depending on how you configure them, because their functionality is very similar. I would guess nearly all tools that can do one can do the other. But if you want to use the right tool the right way, I am telling you, with 100% certainty, you should be looking for an “SMTP client”, “email client”, or “mail user agent” (they mean the same thing). You do not want a relay server, and you do not want an MTA.
What you probably mean by what you’re asking is something that receives mail addressed locally (acting as an MSA or an MDA), and fowards it (important: this is not the same as relaying, because the destination address is changed) to an MSA (therefore acting as a MUA). If something is forwarding mail like this, instead of relaying, and calling itself an MTA, that is not technically correct. But you probably don’t need that, unless you’re using something that won’t let you configure how/where it sends mail. You just need to set up something that provides a binary that acts as an MUA. Then set up your daemons to send mail to your actual email address using that.
I’m speaking from experience because I’ve written both an MTA and an MUA, but if you won’t take it from me, take it from Wikipedia:
An MTA works in the background, while the user usually interacts directly with a mail user agent. One may distinguish initial submission as first passing through an MSA—port 465 (or, for legacy reasons, optionally port 587) is used for communication between an MUA and an MSA, while port 25 is used for communication between MTAs, or from an MSA to an MTA.[5]this distinction is clarified in RFC8314.