Buy one (or ten). They're cheap.
Before I went that route, I'd move to a dual-core ESP32-based board. It's better than ESP8286 in every way. The fact that (by default) the WiFi system runs on Core 0 while Arduino code runs on Core 1 may solve your problem immediately.
There's also the DMA options already suggested. For FastLED with DMA (on ESP32), I've written a DMA SPI-based CPixelLEDController class that hooks seamlessly in to FastLED. See Post #2 in this Thread.
I would also certainly put a logic analyzer on the data line (as I already suggested) to see what's really going on.