Go as a group.
If possible, have your supervisor look over both the original grants and the new ones beforehand. He needs to give specifics. It's really up to the PIs - in this case, your supervisor - as to whether to chase a grant. He is there to facilitate that. I have a suspicion that he may not have wanted to do that particular grant. Some are unwieldy and awful but if he can't be specific about the poor fit, he's bluffing.
His time may be limited, so his power may be to say that he only has time to work on one per person and he recommends you apply for the second. Nevertheless, he should be able to set and stick to a timeline. Granting is all about deadlines and even if he has other grants to do, on deadline, if he is an "expert," he should know fairly well how much time a given grant will take.
Once he sets and you commit to a timeline, you are perfectly within your rights to simply say, this is when I will be able to work on it. I won't be available in the summer so it needs to be done now. Asking your supervisor for help, especially now that his supervisor approached you, is perfectly appropriate, in my book. Especially because John seems to be having issues with the very people he was hired to help.
Other forumites might weigh in. Good luck!
Also, I would reframe yourself away from the anxiety, at least for public purposes. Don't let them pigeonhole you.
Everyone gets upset. Choose a new brand for these purposes - you are a great multitasker but others need to do their part. You get stressed when they don't meet their own deadlines.
Start saying things like, "I'm of course CONCERNED that these funds come in. The whole lab has a great number of things that WE must coordinate. WE need to be able to plan so it is, of course, a CONCERN when others don't stick to the plan. WE're on a tight timeline." Etc.