With a quick bit of desktop research I was able to find this resource. Although not Cape York specific, it does have information on temperature ranges in rainforested areas of the Clarke Ranges in the Mackay region of central north Queensland.
If you’re unable to find more Cape York specific information, I would suggest using data from other regions that data is accessible for, and then make an average differential assumption. For example, the average temperature in summer in the Clarke ranges is 17-27 degrees, whilst the soil temperature according to the resource above is around 20-21 degrees.
I would also suggest looking for data in regions on a similar latitude; the Kakadu National Park in the N.T. spans the same approximate latitudes as the Cape York Peninsula, and there may be data on soil temperatures available for this region.
Remember, it is ok to make some informed assumptions for your solution, as long as you’re able to justify the decisions that led to that assumption. In principle, this cooling idea is a really interesting one, so I’d encourage you to do more research to find Cape-specific data, and if that is not available, to explore these other options outlined above.
Hope this is helpful for you,