Hi Cherry & welcome

!
when I try to ssh/pat him back to sleep he sleeps for about 10 - 15 mins and then wakes either fussing or crying but it gets louder and louder so I try ssh/patting again which works for a short while.
If shh/pat only settle him for 10-15mins (or 1hr even) & he's waking up again, then he could very well be hungry. He's only 10 weeks so 2 night feeds at this age is very normal (if not better than normal

).
Also looking at the routine, he doesn't seem to be sleeping too well/enough during the day. Is he only napping for a total of 2hrs & awake for 3.5hrs before bedtime? I think he is also very overtied which could be another contributing factor to night wakings.
His awake times shoudn't really exceed 1hr20mins (the morning A time is usually shorter so only about 1hr). Here's a link to sample EASY routines for los of your age. The idea is to have 3 long naps & a catnap. Of course its not always possible to extend a nap so you might not have true EASY but still try to space the feeding out so that he is not snacking.
The morning A time is often shorter than other A times & some babies are comfortable to stay awake for just an hour. Try to have him in the cot by the end of A time. So in the morning feed, a little play & then you will almost have to do the windown to get him ready for a nap within an hour of being awake. They tend to get overtired very easily at this age so by the time you see a yawn, he may very well be ready for a nap & get overtired by the time he makes it to the cot. Same for all the other naps. Start your windown an hour after he is awake so that he is already in the cot by 1hr 15-20mins.
Is he actually feeding 30mins non-stop or is that how long it takes between sides?
hth
Layla