Huawei Honor 7 and Oneplus X fit your needs.

There are yet to be launch Samsung Galaxy A9,
Then, OPPO R7 Plus, Lenovo Z2 Pro, Sony Xperia C5 Ultra.

Of course, Mate 8 is the king of 6 inch for now, nearest contender would be Samsung Galaxy A9, third would be OPPO R7 Plus.

Main different would be G8 camera has OIS, which capture less blur image.
Then the processor different, Honor 7 Kirin 935 should be more powerful than G8 Snapdragon 615 processor, but not by much.
Honor 7 front camera is 8MP, while G8 front camera is 5MP.

Both pricing are comparable though.

I would pick G8 for it's bigger size screen.
Both phones' performance should be comparable.

Nope, the Sony Z Ultra, 6.44 inch market sold badly, they focus on Z2, Z3, Z4, Z5 only. Then the Z tablet market.

Closest thing you can get, maybe only the C5.
Older Xperia T2 is below par.

Which ASUS you are picking? Zenfone Go or Zenfone Laser?
There are Alcatel Flash 2, Lenovo Vibe P1m, Meizu M2.

Alcatel Flash 2 is the most balance among these, with 3000mah battery, 13MP camera.
Lenovo Vibe P1m has 4000mah battery.

There is AP set Xiaomi Redmi 3, with 4100 mah battery and 13MP camera, but won't recommend if it is AP set.

Assume all phone are stable, you can ignore the AP set, which has long repair period compare to local original service warranty. Local original warranty are much better, and local original set has better support too, less problem.

Ranking may list like this:

1. Huawei Honor 7
2. Redmi Note 3 (Local Malaysia set)
3. Oneplus X
4. Huawei Honor 6 Plus
5. Lenovo P1(5000mah battery)
6. Huawei Honor 6
7. Huawei Honor 5X
8. Xiaomi Redmi Note 2
9. Meizu MX4 32GB
10. Meizu M2 Note/Meizu M1 Note 32GB
11. Samsung Galaxy J7

These are near or same as Galaxy Note 3 performance in normal usage, but heavy gaming is not better than Galaxy Note 3. Budget phone normally compromise on graphic chip.
You may still play smoothly, but will experience frame drop, or lower quality rendering.
Then, budget phone do compromise on other area like screen quality, additional feature like IR blaster, NFC, speaker loudness, lack of Micro SD slot, lower battery life, worse camera, and build quality too, not just the performance that matter.

The lagginess of phone all lie down to processor performance and software optimization, RAM at least 2GB will be sufficient.

So far, only the original ASUS Zenfone 2 with Intel ATOM z3560 or z3580, Oneplus One,Oneplus X, Xiaomi Mi4, ASUS Padfone S with Snapdragon S801 processor, or some cheaper phone with at least Snapdragon S800 processor able to be better or on par with Galaxy Note 3's Snapdragon S800.

This post has been edited by Alan Yee: Mar 19 2016, 10:45 PM

ASUS Padfone S, Snapdragon 801 processor with Adreno 330 graphic chip confirm better, but it's size is only 5", battery only 2300mah.

If not, another option is ASUS Zenfone 2, Intel Atom 3560/3580 processor with PowerVR 6430 graphic chip.
Bear in mind, not Zenfone 2 Selfie, Zenfone 2 Laser, or Zenfone Max.