maybe start using it yourself, and giving it to your older kids, to use it, too, and all pretend to act happy and calm while it's on?
Can you forego doing it for a day or so, so he can get over the fear a little bit while you do this?
Maybe administer little treats to everyone who uses it? (of course, since your 14-month old is sick, he shouldn't have anything sugary, so give him a healthy snack for a treat)
Maybe when you use it on him while he's sleeping you can just hold it over him and not let it touch him (I'm assuming it looks like an oxygen mask), and just very, very gradually put it on his face. Or maybe it will work well just hovering over his face. Better than nothing, I suppose.
Is the noise scary? At night you could mask it with soem white noise, that you turn up very gradually,, like a fan from lower to higher settings.
Good luck. I know you must feel bad but you're doing the very best you can. Just give that little boy lots and lots of cuddles and see what works best.