You absolutely must not connect your inverter output to your 'main breaker' if you are grid connected.
You should not be using welding cable to connect residential AC circuits, welding cable will only be rated for low voltage and won't have the required mechanical protection of NMB type cable typically used for building circuits in the US (according to
here)
No problem using welding to connect the DC side of your inverter to your batteries, though it is somewhat over-specified and usually more expensive than automotive-spec battery cable.
I am not an electrician but assuming that your inverter outputs single-phase 110V, the inverter is rated to supply 27A so 10/3 NMB cable should be OK. Don't forget to protect the cable with an appropriate MCB (
Miniature Circuit Breaker) and your onward circuits with a GFCI (
Ground Fault Circuit Interrupter).
Don't forget your earth grounding rod and be sure to provide that earth through to all AC outlets and to your inverter enclosure.
Did I mention that you must not connect your inverter output to the grid?
