Lunch and the 2-1 transition

[momma.bear]

My LO is on day 2 of the 2-1 nap transition. Yesterday he fell asleep in the car for about 15 minutes, so our day went like this:

6:45 WU (bf around 5:30)
8:45 breakfast
11:15-11:30 nap in car
12:00 lunch
12:30-2:45 nap

Today, however, went like this
7:00 WU (bf 5:45 one side, 7am other side)
8:15 breakfast
11:15 offered lunch - refused
11:30-1:30 nap
1:30 offered lunch again - refused

My questions:
1) he really struggles with eating breakfast and lunch. Breakfast is usually offered around 8am, but he won't actually eat anything until 9/9:30. Do I offer it at 8am only and then nothing until lunch?
2) he's too tired even by 11:00 to eat lunch, and then when he wakes up, likely because he's still tired as he's just transitioning to one nap, he still doesn't want to eat. What do I do here? He's a very picky eater to begin with, but even his favourite foods are thrown on the floor if he's not interested.

[amayzie]

We often would split lunch- have a snack before the nap and a snack after or something simlilar- it's fine to spread the meals out across the day at this point.. they may be interested in eating little bits at a time rather than a large amount!

[momma.bear]

He won't even eat little bits. Yesterday for breakfast he ate about 3 bites of oatmeal cereal, 2 small pieces of fruit, and then just refused to eat more. However, when I offered him his favourite granola bar he ate that in a hurry. Then at 11:00 we did lunch, but he just threw anything I placed in front of him. So I ended up just letting him play and fed him as he was playing. I know this is bad, but when he is tired, he just gets upset unless he's busy. Sitting still just doesn't happen. I gave him a small snack of cheese and cheddar crackers (food I knew he wouldn't resist), and he ate that no problem. But then 5:00 dinner was pretty much refused, he was in tears in his chair and wanted nothing to do with eating.
Any suggestions?