Well, the obvious and permanent solution is a full gearbox overhaul and how you go about that depends on you. If they are worn, this can include replacement of the plain bearings and reaming to size afterwards as well as routine replacement of ball races. If your workshop doesn't run to the necessary tools then its a matter of finding one that does and I suppose for an Ariel owner the obvious candidate is Draganfly Motorcycles.
As this thread started with you only being concerned about a grinding noise when putting the bike in first gear and the sad story about Heinz bean tins only came out later it would seem sensible to check out the clutch at the same time.
This all sounds as if it could cost a bit but as I said earlier, once done you will never have to worry again and as any 1950 Ariel is a keeper and is capable of any ride you will ever want to do then it is cheap in the long run. I did a lot of foreign touring on on a VB Ariel that also pulled a chair at times. I have good Ariel memories.
Cheers,