Aiyoh this is so draining kan, a few of us went through this phase and it's totally normal at 2 months, their sleep cycle is still super short and they startle easily when you move them! Try keeping baby on your chest a bit longer after feeding (even 10-15 mins) until they're in deeper sleep before transferring, and make sure the cot is warm so the temperature shock doesn't wake them up; some mums I know swear by putting a heating pad on the mattress first then removing it, or even just keeping one hand on baby's chest for a minute after laying them down so they know you're still there. The 4th trimester is honestly brutal but it does get better, hang in there, mama!