On a PAL Amiga, a full line sweep translates to about 454 virtual Lores (Low Resolution) pixels. (64 µs duration of a horizontal line, 7.093790 MHz pixel clock, which is about 140.968 ns per pixel: 64 µs / 140.968 ns ≈ 454).
I say virtual pixels because not all of those pixels are actually usable as there is the horizontal sync period of about 4 µs and the back porch of about 8 µs, leaving about 52 µs or about 369 pixels. In practice, the number of used pixels was even lower (so it would fit into the visible non-distorted area of the CRT) - usually 320 pixels in Low Resolution.
Therefore, the number of pixel-times where nothing is displayed is about 134. Copper lists allowed positioning (waiting) horizontally with 4 pixel granularity and a copper MOVE instruction took 8 pixels to complete. 134 / 8 are a little more than 16. So you can update about 14 to 16 color registers each line outside of the display data fetch when using copper lists.
So assuming we use 5 bitplanes (32 colors), we could use 16 colors for the background and prepare a copper list that changes those 16 colors (or most of them) each line, providing for a very colorful setting. The copper list could be prepared once when switching to a different scene and then not changed anymore while the scene is active, or it could be prepared once per frame during the vertical blanking area.
We could then use the remaining 16 colors (which remain static) for blitting the animated character onto the background.
I checked the game for what it actually does and it pretty much matches:
$6200, which means 6 bitplanes without HAM mode, i. e. EHB (Extra HalfBrite) mode (64 colors with 32 freely controllable palette entries and the remaining 32 colors roughly half as bright as the first 32), as you originally suggested.
Apart from that it uses the technique I outlined above: In the Copper list it does a
WAIT $xxE1FFFC for each raster line and then changes a couple of colors (about 2 to 8 per line) out of the first 16 (leaving index 0 alone - as a consequence it also affects color indices 33 to 47 of the EHB image) and doesn't touch color indices 16 to 31 (consequentially, indices 48 to 63 of the EHB image also remain static).