Hello Rebecca. I have a similar gap to you. DD1 is nearly 6 and DD2 is 5 months.
Definitely worth getting a second opinion from a BF support person. But also, at 6 weeks this does sound very normal.
The way I managed in the early days was to feed DD2 as DD1 got organised for school, this entailed getting out of bed pulling on clothes, brushing teeth and issuing orders from the sofa! Put DD2 in a sling for the school run.
Come home, turn on telly, ensure remote, cakes, flask of tea to hand, along with mobile. Park bum on sofa and feed DD2 until she crashed out. Put DD2 into Moses basket. Replenish flask, attempt to put washing on, clear breakfast dishes, snooze, etc. Continue on this loop until it was time to do school run again.
Having big sister about it entertain DD2 after school allowed dinner to be assembled.
Get DH to sort dinner when home or cuddle DD2 so I could do a few things, like homework and wrestle uniform off DD1.
DD2 then would sleep/feed through dinner.
Once DD1 in bed early evening, bath DD and feed until conks out. Keep in Moses basket before taking up to cot when I went to bed.
It gets easier as space between feeds increases - unless on a bloody growth spurt - and eventually they sleep a bit more in the night - until bastard 4 month sleep regression hits and stays around.
Lower your expectations and if you can, go with the flow for a bit, and you may find you fall into a routine as your LO gets older.