Huge sympathy here, DS is five months and he went through a very similar phase at 12 weeks, where feeding to sleep didn't work for him consistently but he didn't have a plan B.
Firstly, you're probably not "missing a window" - I have days where I track DS's tiredness cues perfectly and others where I do exactly the same thing and it results in hours of wailing...
I'm probably not the best person to advise as DS's sleep went to pot at 12 weeks after sleeping 7 hours at night for the first time, which wasn't frustrating or anything. My understanding is that sleep patterns change a lot around this age as babies develop sleep cycles and learn to fight sleep. I've found posts by FATEdestiny on this board about the inaccurately named "4 month sleep regression" really helpful in understanding what's going on.
I'm keeping a diary and have noticed that we have better nights as a rule when DS has at least 3 hours worth of daytime naps; he generally gets ratty anything from an hour to two hours after waking.
DS will now usually feed to sleep again so, for us, that was a phase. For daytime naps, I try whatever I think will work and try to time it for 90 minutes to two hours after he's woken - this morning was a feed and snooze on the bed as we had a very similar night to SkiptonLass2, yesterday's afternoon nap was a pushchair stroll. I must admit, I find it easier when we're out for much of the day and try to plan outings around naps where possible.
It's so frustrating, isn't it?! Other tricks in our arsenal include dummy, rocking and sling. None work consistently but all have been known to work...I definitely find its best to just roll with it, but it's easier said than done on days where I'm so tired I feel sick. I find
and
help immensely, though.