I would suggest going back to basics and making sure that your current configuration is as good as possible first. Things change; neighbors buy/change network equipment continuously, gov't, city and utility companies install new equipment as needed.
After doing all the above and the situation doesn't change, I would be inclined to think that your 3+ year old pair of Bridge/AP's may not be up to this demanding task anymore.
If you're streaming on the 5GHz band; I would recommend an AC1900 class router or two (more distance, more throughput and more horsepower vs. either the Bridge/AP's or the RT-N66U you have now.
If you're streaming on the 2.4GHz band; I would recommend another RT-N66U with both flashed to the 374.39_0-em RMerlin firmware. And follow the above (link) to optimize your network as much as possible.
As for the Netgear EX6200, since it is not available or even tested yet - that would be hard to recommend one way or another. I would be leaning to forget about Range Extenders though - there is nothing about them that suggest they work better than an additional router switched to Bridge mode and are not as reusable either (i.e. too focused in their use, or too limited in their capabilities other than that one specific task).
I'm confused, is there just 1x WNHD3004 or 2? You need 2 to benefit from it's HD streaming abilities. If you only have the one, I would suggest either a second WNHD3004 or 2x AC1750 or greater routers with one setup as a bridge, just as L&LD had suggested.
The EX6200 is only 2x spatial streams(same as the WNHD3004), while an AC1750 or greater router provides 3x spatial streams. The only benefit the EX6200 would provide is the 80MHz channel width(would require upgrading to a AC1200 or greater router to take full advantage of it) vs 40MHz channel width of the WNHD3004, which is much less important in terms of performance than the extra spatial stream as well as channel width you gain by getting two AC1750 routers and using one as a bridge.
