I'm a single mum to my DS (now 2) but went through exactly this myself so I know how exhausted you must be feeling right now.
I was at my wit's end when he got to be around 10 months although I was still committed to BFing & in desperation asked advice from some of my more experienced Mum friends.
1 who has 5 of her own invited me to stay the night at hers & promised me it would be sorted by morning. I wasn't convinced but was so tired I had to give it a go.
I bought a goat milk formula & a set of Tommee Tippee bottles from the Closer To Nature range (the teats are apparently more similar to our nipples than any other brand/type, which to look at they certainly weren't but they must feel ok as he did eventually latch on).
So we went over to her place & when it came to bedtime we gave him a nice relaxing lavender bath together (I mean we were both in the room with him not actually in the tub! ), got his sleepsuit on & then my friend said "Mama's going for a sleep now, I'll give you your milky!". She'd already told me not to make a big fuss of him before going upstairs so I gave him a big hug/kiss & said goodnight before leaving him with my friend.
I lay down in her bed (she'd said they'd sleep downstairs on her sofabed) & could hear him screaming like he was being murdered, interspersed with soothing sounds & reassuring talk from my friend, trying to calm him down & every now & then offering him the bottle of warmed goats milk. This went on for what seemed like ages & I was just about ready to give up & go down to him but I was so shattered I must have fallen asleep before I did because the next thing I knew it was morning & I'd slept soundly/uninterrupted for over 12 hours in my clothes!
When I went downstairs my son was playing happliy with my friend & she told me that although he was very resistant to taking the bottle, he did eventually drink a bit (not a full bottle but enough to see him through the night).
I was a bit apprehensive taking him home that evening as I was supposed to do the same thing myself & wasn't sure he'd do it for me but I got myself into a more positive frame of mind before I made up the bottle & made sure I kept my breasts completely covered. To my amazement although he did ask for breastmilk at first he took the bottle when I said there was none there anymore at night time but that he could have the goaty milk instead. The bedtime feed was warmed but if he woke up in the night for more I always gave him it cold so there was less incentive to wake up for it.
The only times I lapsed with this was on the odd occasion when he was ill & I allowed nightime BFing but he accepted that it was back to the bedtime bottle once he was better.
We carried on BFing during the day til he was 18 months old, which for me was just right. I felt I'd done my bit & had enjoyed the closeness it brought but wanted my body back & by this time I could talk things through with him a bit more, so with a few weeks preparation the complete weaning off the breast was gentle, calm & stress-free for both of us.
I'd say you have to be clear what you're doing & be very determined or it won't work. Also, follow your instincts, after all everyone is different.
Best of luck!
Hope this helps!
xxx