I would try a cup which requires no compression and has no valve, the hard spout type, if you tip it the water comes out. Or a straw cup where he can suck instead of tip the cup. You don't need to buy a straw cup to try out a straw, get a wide type straw and just use that in a regular cup whilst you see if he likes it (you can hold the cup to prevent spills). With a straw you can trap some water inside by dipping in the water and putting your finger over the other end, a little will stay in the straw which you can put to LOs mouth so he gets the idea it has water in it.
If the straw is successful you could look for the type with a lid and straw which daycare may accept.
Otherwise, he may just not be thirsty. He may be getting the fluids he needs from the 3 bottles and whatever solids he is eating. If he has purees, soups, stews or juicy fruits (oranges, grapes etc) these have quite a lot of fluid anyway.
Mine didn't drink large quantities of water either, just sips really. I was amazed even when he was 2.5yo he would still have only small drinks whilst his friend of the same age would down a half pint of juice - of course juice is sweet so drinking larger amounts isn't necessarily from thirst but from enjoying the taste - I'm not suggesting juice btw only that the difference between LOs is quite vast.
Just a reminder, only put an oz or 2 in the cup so it is not heavy.
He might be more willing when a bottle is switched to a solids snack with water which isn't far off I expect.