On a Sunday, no issue with taking bags on the tube
It's a pretty easy walk from the Hatton Cross HGI to the station, but IIRC no lifts so you'd need to carry your bags down 2 flights of stairs
If you can't cope with stairs, you can always take a local bus for free (it's all in the Heathrow free travel area) to T4, then get the tube there as it's step free at the terminals. Would probably add 15 mins to the journey