The feed mechanism of the 1919 depends on the overall length of the cartridge for reliable feeding. Cartridges that are too long will not fit on the feed tray and ones that are too short will not be picked up by the extractor. Also, soft-point bullets will be driven forward by the extractor during the feed cycle and won't get picked up. You need a cartridge that has a full-metal-jacketed bullet that matches the overall length of GI ball ammo. Old surplus is getting very hard to find, so you will probably wind up paying for modern ammo that matches the old specs like PMP or PPU. Personally, I would avoid steel-cased ammo, but it would probably work.
Right now on Ammoseek, Turkish MKE .30-06 is being offered for around $ .73 - 77/round. I know nothing about this ammo, but if it is of good quality, it would work well in your Browning.