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I guess the question is how much 2x4 can you fit into a large cup (Im sure someone here knows the answer to that), you can also buy a box of 2x4 from LEGO stores I think depdning on what they hav for their PaB (im sure why would not mind if you did as well) it is a bit pricey, but if you need a ton it still may be more cost effective as you will not have to pay shipping on it either. and then seeing what shipping and cost would be on a site like Bricklink or Brickowl.
the other option is from Parts direct but price per price will probably be kinda high BUT the shipping is a flat 2.95 (at least in the US)
On a related note, I haven't asked for a box at the LEGO store in some time, but I have heard you can no longer buy by the box, yet I keep hearing people recommending it and saying you can... any confirmation either way? Is it up to the store managers?
LegoLand Discovery centers, which are not owned by LEGO, have that gimmick, and with the pricing I saw you want definitely want to buy by cup and not weight
As I've no idea what you are using them for thought if throw that out there.
I should add I strongly recommend buying branded lego but since you did ask for cheap bricks it might be ok for a one of kids event or filler.
Now with a Box of 2x4s from a LEGO store, you are paying sales tax likely (US wise anyway) but still may be cheaper than what someone may charge to shipping you large quantities of used brick (especially because they are 2x4 and can get bulky if someone is not stacking them)
But that said, a box of 640 2x4 from Bricklink would probably be about the same size as a box of 640 2x4 from TLS. I'm fairly sure of this ;-)
The weight, going from Bricklink, will be 2.32g x 640, which gives 1.484kg. I don't know what postage is like in the US, but you could get that package collected from your door and couriered UK to UK for around £10. Factor in another £5 for packaging, and that's £30 for 640, versus about £60 + transport for 640 from TLS (although we don't have that option in the UK).