So to correct the other answer you got: it depends on what you mean by "start right with the DLC", I suppose. More specifically, if you intend to "start right" on launch day this week.
They don't support save transfer from original PS4 game -> Remastered on PS5 at launch, is the thing. However, there is a patch coming in (tentatively) 2-3 weeks that will add that. So if you're fine with waiting that long, you'll be able to transfer it then; note that you may still need access to the original PS4 game to export the save, depending on how it's implemented.
That being said, if you don't want to wait I think you can still play the DLC in Remastered without having to replay the game, but you might still have to play the intro of the game first and you might be without many upgrades due to how early you are? I went searching for specifics on that last part, but what I found wasn't very clear. It seems that you can switch to the DLC campaigns from the menu when you're out in the city, regardless of story progress, but it's not directly from the title screen menu or anything like that, and that's why you need to do the intro first.
I'm in the same situation and will be waiting till the patch to play it, as it'd save me $5 compared to just buying the DLC on PS4 and I do want my save to be there. It is kinda silly to wait that long and not have a non-Astro/Bugsnax PS5 game for that time, but... I have other stuff to play anyway, I can wait a little longer for Miles.
