Suggest you try Gloster Air Parts; Pete and Diana have most of the regular RV hardware bits in stock. LAS Aero are the other port of call.
Most builders have several left over after construction, so you may find some offerings from someone's 'bits box'.

The Cherry MSP-43 would be an acceptable substitute. It's a bit higher grade; same steel shank, but a Monel rather than an aluminium head with better corrosion resistance and strength.  
 
 Or find an RV-12 builder who wouldn't miss the odd hundred LP4-3s!

So as to complete the thread and act as guidance for anyone else who needs some more LP4-3 pop rivets:
I have sought and costed out the alternate pop rivets in the UK and found that, even with P&P from the US, the Vans pop rivets are far more cost effective at 5c each so have placed an order through the website. However, I am sure that Gloster Air Parts would have been competitively priced had they had some in stock.
 
Moderators: I'm happy for you to delete the thread/post or leave it up for information for others.
 
Cheers