The hills are further than 5 miles no?
I haven't been there but it appears a 750'+ hill at 5 miles. The 4,000' MSA contour on the chart I've got starts at 9nm in the direction of the 330 radial.
There are several reasons why I think it is an unrestricted climb to 2,700': a) because the chart(s) effectively say that, b) you can't set 1250' for a go-around on your average airliner AFDS (MCP/FCU or whatever) and c) because it's much more sensible to do it that way.
LTAI is a place I go within range of on the way to/from the Middle and Far East so I might need to divert there one day. I've seen nothing in our documentation (Navtech) that precludes direct climbs to the final altitude on the missed approaches and that's the way I'd fly it if I had to do it tomorrow...