Bluetooth Speaker Question

[FocusST]

Hey everyone, I've got an interesting issue I've run into and hopefully someone here has some experience in this area, as it's kind of an odd setup that I've got. So I recently installed a 1000w Pure Sine Wave Power Inverter in my '16 Ford Focus ST and it's powering my Sony GTK-XP7 Bluetooth Speaker to give my car audio some extra mid range and bass, as well as providing lighting effects. It's kind of silly but I thought it would be an interesting project. Anyways, I've ran into the same problem that I had with the last car I installed it in: it lacks bass/power. When I use it indoors off a standard 120v outlet it plays just fine, with lots of bass and full sound. But in my car it just sounds weak. I don't think it's the car battery because I replaced it in my last car and it didn't fix the issue. I wasn't sure where to ask this question as it's car and electrical related so I thought I'd start here. Here are some stats on the bluetooth speaker:

Power Output = 260 w

RMS Output power

Left/Right Channel = 470 w

Power Consumption = 60 w

So looking at the numbers the power inverter should have no problem powering the speaker. I'm thinking either I need a better car alternator possibly or there's something else I missed. Total wire length is about 10' (trunk to engine bay). I used 2 gauge wire from the power inverter to the fuse and then 4 gauge from the fuse to the battery (a little over a foot in length). I realize I could of used a thinner gauge wire and been fine. I also realize I probably should of stuck with the same gauge wire from start to finish. I will probably fix both of these issues once I figure the lack of power dilemma.

Side note: I used a modified sine wave inverter before my current one and it had the same issue. Not sure if that helps at all but thought I'd mention it.