My youngest lived in a sling for the first few weeks and months (and fed in there most days), so perhaps that would help? That way, baby gets cuddles and good, and you can still do things with your toddler, go to the park etc.
Realistically, it's not viable to never pick up your toddler for 6w, but just do what your body is comfortable doing, and as safely as possible. I started off at about 2w lifting from standing, so was able to put my 2yo on the swings in the park, lift into the car etc, but I didn't walk whilst carrying until about 3w pp (and only in a sling in my back, so minimal twisting, spread weight etc). I'd your body isn't happy (it's not worth straining yourself over) then out a pillow on your lap and have him climb on whilst you're sat down.
There will be times they everybody cries, been there, it's rubbish,but they do eventually pass.
Don't be afraid to let your toddler watch too much TV, but also, if you can make it out to the park, go for a stroll etc then it's likely to help your toddler feel happier and aid your recovery.