Obviously you need to follow what you think is right based on the information available and advise from your ped. I will say though that when my LO was taking very little milk in bottles (he always took half the guidance amount) the advise I was given (health visitors, GP, BWers, books, internet etc etc) was ALL to increase the dairy solids he was offered. Well I tried that and the more dairy solids he ate the less milk he took. Being under 6 months old I knew milk was still his main source of food so in a radical experiment I cut out all dairy solids and guess what, his milk increased (and I was happy with that as the formula was vit fortified where as the cheese and other dairy foods were not). I'm just saying sometimes the advise can guide us the wrong way. I'd say if your LO isn't taking enough solids you need to cut back on milk, not keep it going with it and driving yourself crazy trying to transition milk to a sippy.
I honestly think you are confusing the issue with having two different and (almost) opposing goals that you are tackling at once. If you want him to keep the milk intake up keep the bottle, just keep it, the physical action of drinking from a bottle is not harming him in the immediate future (on it all day or for ever yes, will damage his teeth but that's not the case here, it's a quick bottle, if anything the BT bottle posses more 'threat' to teeth if they are not being brushed before sleep and this is a real problem that I would certainly tackle if you haven't already). Switching a bottle to a cup for milk mid day is not at all the same as switching or dropping morning or BT milk. That's not how others drop bottles because LO WILL throw a fit if they want the milk or want the bottle. People drop that mid day bottle by offering a snack and sippy (of water or milk) at that time with the acceptance that milk intake WILL drop (because LOs don't want milk in a cup and because it's fine for milk intake to drop at a suitable age, usually before 12 months).
In all honesty I think you've given yourself a bit of a pointless goal in trying to switch that bottle, even successfully getting him to drink a full amount of milk from a cup at that time of day will have no effect whatsoever on his dropping the morning or BT bottle. Every bottle is dropped separately and will have it's own weaning process to go through. Mid afternoon bottle switching to a cup is not a 'practice run' for dropping the morning bottle. Hope that makes sense.
(as an aside I'd drop the afternoon bottle and then the BT bottle and keep the morning bottle. Very many people keep the BT bottle so that sleep is not disturbed by a routine change but there are benefits to dropping that one earlier and maintaining the big milk drink (bottle or cup) in the morning. Bt bottle can be weaned in 2 weeks or less with no crying and no sleep loss)
I'm hesitant to offer water in a bottle because I'm also trying to drop that bottle all together
Well, it's part of a process. If you need the milk to stay up then just give the bottle and there are no tantrum issues. When he's had his dairy trial or at whatever point you are ready for the formula intake to drop you can attend to the habit. If my LO really NEEDED that amount of milk I wouldn't be trying to switch the bottle, my primary goal would be to offer him the food he was desperate for in whatever form he would accept. The point Laura and I have been making is that we don't think he needs it, but if he does, really does, for his health then why would you risk it reducing?
My suggestion of offering water is one step in a weaning process. Once he worked out he could get the pleasure of sucking but only water he will increase his solids intake which is what you said in your original post was an issue (holding out for the bottle, not eating lunch). The bottle of water would satisfy his need to suck but not give him the calories, so he would be hungry for solids. And a short while later he wouldn't bother with the bottle because he wouldn't be hungry for the milk, wouldn't be getting the milk and would switch easily to a sippy of water - also because he already happily takes from a sippy and because he would be full from solids.
It's about going one step at a time - you can go even slower on a bottle wean to totally avoid any crying or tantrum...if that's what you want. But not if his milk intake needs to stay so high.
At the risk of being repetitive, too high a milk intake can lead to an underweight child, they are too full on milk to take in enough solids, and the milk doesn't offer the same level of nutrition as they need (certainly iron deficiency as Laura mentioned already).