In C18 and C19, iron was the only metal to be consistently cast in sand. Copper, lead, tin and zinc were consistently cast into cast iron moulds (originally stone moulds). The primary reason for this practice was not to contaminate the non-ferrous metals with sand. It took a lot of fuel to refine them to the levels of purity demanded by the trades and industries that consumed these metals so there was every incentive not to introduce contaminants when casting them to ingots after the final refning. Indeed, these metals were usually priced in proportion to their purity. Iron in comarison, was of lower value, was produced in much greater volumes, making simple sand casting in open beds convenient, and the metal naturally contained a proportion of silca anyway, making modest additional contaminatio n of little importance. Ultimately, iron too came to be cast into cast iron moulds (although sand pig beds lingered until as late as after WW2 in some works, albeit with a commensurate reduction in prices acheived by the owners).
Each non-ferrous metal had long established sizes and shapes of ingots that saw minimal change until the late C19, with most of these traditional ingots continuing in use well into C20 for more conservative markets. For specialist markets (notably for trade to non-industrialised societies), the various metals were frequently produced in shapes and sizes that very closely met the traditional preferences of those markets. The small ingots demanded by the North American fur trade are a typcial example.
The traditional shape and size for pig lead was around 3ft long by around 4in to 6in wide by around 4in thick. The ends of the pigs could be squareish or half-round. Weights varied from around 2cwt upwards.
I have not encountered references to casting pig lead in any other shapes / dimensions (although I have only consulted a couple of smelters' accounts so my knowledge of this aspect from primary sources is very limited). My belief is that lead was exported in normal size (3ft) pigs. I presume that that larger fur trading companies bought pig lead direct from UK smelters / non-ferrous metal merchants, but that smaller operators may have bought smaller quantities from North American merchants who imported cargoes for general sale and sold-on to manufacturers, plumbers, glaziers, fur traders. etc. 3ft ingots wouold have been inconveniently large and so either the North American merchants or other, lower levels of middle-men, recast the lead to smaller ingots suitable for the fur trade. Does this accord with the information you have so far assembled?
Fur trade company accounts are the most likely place to find written records of the dimensions of these smaller ingots. The big companies were very exact in the specifications they issued, knowing very clearly the preferences of their markets. There has been a lot of published research into the activities of the NW Co, the American Co and the Hudsons Bay Co. Chasing up published references may well lead you written specifications, e.g in orders from trade posts to HQ, or in invitations issued by company HQs for smelters / non-ferrous metal merchaants to tender for the supply of certain tonnages of trade goods in specified sizes and qualities.
I concur with your belief that the recasting to small ingots probably mostly or entirely occurred in North America. However the possibility exists that the recasting (or, rather, the lead may have been directly cast to small trade ingots after its refining) may have occurred in the UK. I raise this possibility because the UK copper smelting industry specialised in casting a number of small and unusual dimensions of ingots and related products for sale to companies trading to non-industrialised societies. However, with lead being so very much easier to recast than copper (which requires a much higher temperature and needs skill and care to achieve the correct 'pitch'), perhaps a comparison between the two metals is irrelevant.
The answer will more likely be found in fur companies' records, I suspect, than in UK smelters' records.
Do post the results of your research!