My 1 year old afternoon nap is becoming too close to bedtime?

Hi all mums and mommies, I am a first time mummy so hence why i ask for your advice. My little gal has just turned 1 and she normally wakes up about 6am-6.30am and by 10-11am she is tired so i put her down.. she will sleep about 1- 1.5 hours. She has her lunch after this and by 3-4 pm which is tired again and sometimes will sleep up to 2 hours hence she is waking up 5pm or 6pm. So then i am putting her to bed by 8pm and she is not ready and will fuss,play some whinging and will go to sleep 9pm-9.30pm. I am guessing she is ready for a 1 afternoon nap? Should i try and skip the morning nap even though she is tired, irritable... just push it for 12- 1pm nap. I tried a few times and she will wont sleep till 3pm being more upset overtired and sometimes just sleep the 1-2 hours and still wont go down at 7pm bed time. So then she has had less sleep. I would also like to point out she sleeps only about 9 hours a night now she has turned 1 year old.. she use to sleep 11-12 hours thru the night but since this nap time mess she is only sleeping 9 hours with some random wakes up in the middle of the night.. i was thinking teething was causing this....Some sound advice would be greatly appreciated. Thankyou