It's a low priority "fun" project, so I'm in no rush. I already have several better scopes.
Better scopes ?! Lucky you, I envy you !

This TDS 544A is my beast, spent 500 Euros for it (that's roughly at least as many US dollars), which was a real bargain considering it was recapped and the seller was willing to prove that it actually worked. That's extremely rare.
My dream scope would still be a TDS, but the later 'D' series, so as not to suffer from leaky caps, and have better analog BW and sampling rate. Would love a TDS 784D or 794D, with a memory option, but 700D models are mega rare over here, and when they show up it's usually anywhere between 1500 to 4000 bucks, with just a pic of the unit powered off, laying outside on a wet ground, text saying just "very good condition", take it or leave it ! LOL
Yeah right....

I think once I will be in a market for one of those, I will buy from Ebay US... shipping will be silly expensive but it will still end up vastly cheaper than buying one locally, and I would not have to wait 10 years to find one, and I would be able probably to find a cheap one for repair,, instead of paying 1500/4000 for a unit that's described as "in great condition !", like it were some piece of furniture or something, but which in reality is most likely faulty !!!
Anyway, played some more with GPIB, my dodgy GPIB cable was kind enough to behave for a couple hours, so could exploit all the good stuff I found in the "Programmer Manual" I just downloaded. several hundred pages thick, and all pertaining to my 544A at that (all commands are common to all the 'A' series it appears, great), but none the less an easy/quick read : just a dozen pages at the start to explain how the syntax works, then just a long list/catalog detailing every available command, nicely sorted by type. So I could quickly find the commands of particular interest, and played with them all.
Of particular interest : the scope has an "Application" menu, a dedicated button on the front panel. On my scope this menu is completely empty. Kinda annoyed me at first, I thought the Firmware and/or NVRAM might be screwed in some way, and I might be missing some cool features, how frustrating !
But it turns out that no, I am not missing missing anything, it's normal for this menu to be empty. This menu is meant to work with GPIB only. It's only purpose in life is to offer a way for the operator/user to interact with the GPIB setup : you can create custom labels/menu entries in this menu, then the operator presses the soft buttons to select them, the scope then tells the GPIB controller/computer which button was pressed, and the computer can then run a GPIB program/ to make the scope perform such or such test procedure. So I played with that, see below

You can also display a message window with you own text in it, and you can customize the width/height/position of the window as you desire, can even be full screen if you like.
Now the bad news : this Programmer Manual as thick/comprehensive as it is, does NOT tell you what's the command to clear that bloody err log ! Grrr.... quite disappointed, I am.

You can run diag procedures, and retrieve the results, just like you can do from the front panel, but you can not clear the error log, nor even simply read it actually.
So, I am kinda annoyed that I can't get working this little dedicated DOS utility you gave me Andy ! Still, it's only 10KB so I ha d a go at it. Looked at tit in a hex file editor, thinking all it does is probably sending a single command over GPIB, and since they are all plain/legible ASCII text, I should logically be able to see it in the executable file ! Well... yes and no ! As you can see below, one can see an interesting " sendcmd{ " string ! So the GPIB command must folliow the opening "curly bracket", mustn't it ! Sadly no ! As you can see, all that follow the opening bracket is just a long string ofr random characters, most of which are not even printable ones ! And we can't even see the matching closing bracket ! So... could it be that the text string containing the command name and arguments if any, has been scrambled/encrypted in some way, for security purposes, to avoid simple hacks like what I was trying to do ? Could Tek have been THIS paranoid ?!

Can'"t believe it, there must be some other explanation.... just don't know what it is, yet.
So Andy, you said you might have another software that might be able to clear this bloody error log ? I am all ears !

Anyway, making progress on GPIB then, this first approach was a success I think ! And I am glad because it might be put to good use real soon : might help me repair my TDS 310 !!! I mean, it's front panel is become totally unresponsive but the scope is otherwise working. So GPIB would be ideal to let me take full control of the scope despite the lack of front panel ! But.... that's assuming this scope does have GPIB ! That was hardly a given, considering it was a TDS 300 series hence bottom of the TDS range, and to top it all, the TDS 310 I have is, I think, possibly, the lowest spec 300 one can possibly think of !!! I mean, has only 2 channels, only 50MHz of analog BW and a mere 200MS/s, almost a toy in other words ! LOL But well, I looked at its butt just in case, and what do you know, it DOES have a GPIB port, YEAH !!!! Even has serial and centronics ports as well. I am over the moon. Can't wait to resume work on this repair.