GH4 with 14-140 & 20mm vs FZ1000 / LX100

Ok, you've probably realised from my question that I'm not a pro, anyway.....

My cameras that I've learnt on for the past few years has been the LX5 & LX5, and I want to now make a significant upgrade.

Many people have extolled the virtues of the FZ1000; having a great lens, etc, but it's quite large to carry around all the time, so I would need a smaller option as well, probably the LX100.

On the other hand, the GH4 with a 14-140mm and say a 20mm for street photography would be another option, albeit more expensive.

GH4 +14-140 £1650 20mm £272 = £1922 ouch!

FZ1000 £750 LX100 £800 =£1550.

Maybe the RX100 iii instead of the LX100 which would bring the total down to £1450. (Didn't care for the user interface of the RX100 iii but the results make it worth putting up with.

I need the zoom as my son is involve in some stage performances; dance, acting, etc.

I'm thinking that the GH4 would be the best bet, but maybe the quality of the lens on the FZ1000 & LX100 would make up for the smaller sensor. I haven't the budget or desire to spend anymore on lens for the GH4, especially as these cameras will generate any income.

I guess I could go for the GH3 instead, but I'd quite like to get 4K, especially for the 8 meg picture options. GH3 + 14-140 would be about £1100, and with the 20mm, £1372 - cheaper than the LX & FZ.

So, just based on image quality, etc, would the GH4 be that much better than the two FLCs?