There are a few options.
Firstly, when the bike was struggling in traffic, was the fan working? If you are not sure, check it out before you go any further.
Secondly, I assume you have checked the coolant level. If not, then doh!
Thirdly, does the bike have a full BMW service history? It doesn't mean the previous owner/s haven't been meddling, but it is not unknown for meddlers to not bother with the vacuum filling technique which is vital for the bike. It is NOT a nice to have. If it has been filled without the vacuum kit then the airlocks in the head could be causing the overheating and this will eventually lead to the head cracking. There is no way of checking the vacuum stuff without actually doing it. If it has a full dealer history this is probably OK(?
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The rad could be clogged with shite thrown up by the front tyre, (a really dodgy design fault) or by the fins being mangled.
The clogging is NOT inside the rad. but rather is muck between the fins in the lower, outer corners of the rad. The first check is simple enough, just take a close look at the front of the rad to see if the fins are OK. Lots of bent or crushed fins could well be the cause. If so, careful 'combing' should see the issue sorted.
If all the fins look OK, you need to go deeper. With the side panels, belly pan and fan off and with the help of a mirror and torch, look through the lower corners of the rad. to see the extent of the clogging. If you have a completely clear rad. the problem lies elsewhere. If it extends a couple of inches up and inwards on each corner then un-picking the mess is certainly possible.
Soaking and waiting is the order of the day; mad poking isn't. So, allowing a good half day, wet the clogged area from both front and back and leave it to soak for ten minutes or so. Repeat. Repeat. Repeat! Once the muck is well soaked, try a garden hose with a multi-head jet thingy to direct some sharp needles of water into the clogged area, again from front and back. If nothing substantial comes loose then try more soaking. If stuff does start to wash out, you are in luck.
At this point I tend to use the tip of a very small tie-wrap to poke and prod at the muck, both front and rear. Then more washing to see if more stuff gets dislodged. If more stuff comes out then just continue carefully.
However, the really bad news is that the aluminium core of the rad starts to corrode if the bike is ridden in the winter and if the salt gets into the clogging it tends to turn to concrete in which case it is doomsville.
No one makes a replacement rad. I found a place oop north who would copy my alloy and plastic rad in steel and copper but the cost was as high a new one and although it would last a lifetime it would weigh as much as the rest of the bike.
So, hope for the best but prepare for the worst