There are coolers which are fixed by some kind of spring that gets hold under the left behind posts (sorry, don't have a picture at hand atm).
However, they are a bit hard to find nowadays since this technique was more common for 486 stuff rather than Socket 7.
You could try to make one such spring for this cooler out of some 1.2mm spring steel wire.
It will be some try and error until you get the right dimension and tension but it can be done.

I personally wouldn't recommend changing the socket at all because it is very likely you will damage the board without appropriate tools and skills.
The socket has many ground and vcc pins which are attached to their respective planes which theirselves cover a large amount of the board area.
Without preheating the whole board it will be very difficult to remove the old socket. A hot air station alone will not be enough imho.

That was my case, too. Search for this cooler clip:

You could use one of the clips that clips on to both tabs (usually they have 3 holes so they can go on either way actually) as mentioned before. Some, but not all, Socket 370 coolers come with these. I'd probably bend it a bit to reduce mounting pressure and strain on the remaining tab.

There are coolers that clipped directly onto the CPU. These are less common, but my childhood Pentium machine had one and I did find a second one at some point. The ones I've had were very cheap though, the bearings in the fans are all shot now and you can't really replace the fan since that's what the clips are molded into. But if you can find one in decent condition it would work. I wouldn't use one

Finally, there are Socket 7 CPUs that had permanently affixed coolers. I think at least in some cases this was done from the factory. Pentium Overdrive always had permanent heatsinks, but I've seen normal Pentiums with similar coolers as well. If you could find one of those it would sidestep the problem entirely.

I think there are glues for plastic that 'dissolve' the material on both sides so that the two pieces 'merge' together.
An alternative would be using a cooler that clamps directly on the CPU like those for 486 CPUs. Maybe that exists for Socket 7 CPUs as well?

Edit:

XCVG wrote on 2023-07-14, 22:07:

There are coolers that clipped directly onto the CPU. These are less common, but my childhood Pentium machine had one and I did find a second one at some point.

That's what I meant. Wrote my answer before I noticed that sentence.

HanSolo wrote on 2023-07-14, 22:57:

I think there are glues for plastic that 'dissolve' the material on both sides so that the two pieces 'merge' together.

I'm afraid the plastic is already degraded by age and the glue won't last. The clip handles show wear and tear, caused just by putting the clip on:

I don't think it's a good idea to attach the heatsink to the remaining pin on the side. The 'angular force' (don't know if it's called like that) will stress it too much. And I doubt the heatsink sits really flat on the CPU that way.

Take a look at the photo in this thread. On the top left are heatsinks that are mounted directly on the CPU.
Maybe such a system could be 3D-printed. Here's an image I found with a similar mount to give you an inspiration.

You could also try the old hack of thermal paste with acrylic super glue. Big blob of thermal paste on the center and 4 small blobs of glue on each corner that makes contact with the CPU. Hold it in place for a few seconds and you're good to go. It hold surprisingly well and it's not that permanent, with a small twist it can be snapped out again. Only thing is that you need to mount the cooler in a way that it doesn't prevent the Socket Lever to go up and down, just in case you want to change the CPU freely.

One more idea is to take that part of a socket 370 from a defective motherboard, much easier to find nowadays than a socket 7 as I did.
Heatsink glue for Pentium 233MMX?

Edit: Your board MS-5148 (https://theretroweb.com/motherboards/s/msi-ms-5148-tx2) is MSI not ASUS. I have the same board 😀