I have ported Replay Mod to another platform, and it involved changing quite a lot of the code to remove dependency on Forge API.
Porting alone may be different, but I actually fixed a few bugs (too hacky to get merged into Replay Mod, unfortunately), and even went as far as to change the recording indicator's icon, and allow it to be moved.
The code is available on GitHub under the same license (at least, all the code that is modified from Replay Mod, the rest of the code is MIT).

Replay Mod does not specify the license of the name and logo, and I do not want to violate it.

Licenses do not apply to the name / logo. A trademark would but we don't have one registered because that's not trivial to do and there's not really been any reason so far to do it.

As long as you make clear, whether from context or content, that your ReplayMod fork is merely based on but not the same as this one, you should be good. E.g. calling it "ReplayMod Sol Edition", adding a tiny version of your logo to the RM logo and making sure people know to get support from you if they cannot reproduce the issue with the original.

We'll let you know should this ever change.

Thank you for the clear response.

I have tried my best to follow the changes you requested, but I am going to tell you what I did, so that you can tell me if it is not enough.

I didn't use the name "ReplayMod Sol Edition", as that would look a bit weird in the mod list. Instead I changed the name "Replay Mod", to just "Replay", as it was pretty clear that it was a mod, and sounds less like a "brand" (even if it's not registered) . I changed "ReplayMod Settings" to "Replay Mod Settings".

Because this alone probably isn't enough, I stated in the UI that it was "Modified from the original ReplayMod". I couldn't really add much more, as I was working within quite a small space to display text, but I did add a more comprehensive (including credit to the main authors of this project) note in the README file, asking users to test on the Forge version before doing a quick search and reporting on this repository.

Instead of modifying the original logo (which wouldn't have shown up very well, considering that I am terrible at design), I just replaced it with a film camera icon.

As you can probably tell, I did take some inspiration from Badlion, and even if I may not have entirely succeded, I certainly did a slightly better job (Badlion doesn't mention the original authors on their website, despite using the same file format). d'oh.

I know that this may be considered getting support with a fork, but I am only doing this since it is probably in your interest that you do not appear affiliated with my project.