As a test, try boosting system voltage by having the engine running or the converter on and plugged in while the inverter tries to power only the coffee maker. Not sure how hard this would be to achieve in your rig.
The wire insulation usually is not ridiculously thick, unless one is buying drug store 10 awg jumper cables. What is its diameter and length from battery to inverter?
While there are a few brands which try and actually make a higher quality 12v flooded rv marine 'deep cycle' battery, they are not sold at a tire store, and are quite expensive compared to a wally world/tire store marine battery that will have the same general weight capacity and CCA.
Regarding 12v flooded 'deep cycle' batteries go, most seem to be exceptional junk, that retain poor voltage under load, and then require rather ridiculously high absorption voltages held for a ridiculous amount of time before specific gravity maxes out.
If they do not get this then they are doomed to seriously underperform during their brief lifespan.
Even the best 12v flooded 'deep cycles' made by trojan, are rated at only have the 'lab cycles' of their golf cart batteries.
A lab cycle is a 50% discharge followed immediately by a 100% recharge, which no batttery in actual use by an RVer is ever going to get.
That said a pair os 12v marine batteries should power a 1000 watt load through an inverter much better than a pair of 6v gc-2 golf cart batteries.
