This is not a legal use of Tenser's floating disk
I'm sorry but I don't think this exact tactic should be possible.
1. You can only create the disc at a height of 3 feet above the ground.
Tenser's floating disk says:
This spell creates a circular, horizontal plane of force, 3 feet in diameter and 1 inch thick, that floats 3 feet above the ground in an unoccupied space of your choice that you can see within range.
Unless you are fighting mice, floating just 3 feet above the ground is unlikely to give you any distinct mechanical advantage over enemy combatatants.
2. You can't get it to follow you up the tree
If you create the disk at 3 feet above the ground, it wouldn't follow you vertically up the tree (even if you could climb 20 feet higher up the tree than the height at which you wanted the disc to subsequently hover). See the related question: Would a Tenser's Floating Disk vertically follow its caster?
3. The top of a tree cannot normally be considered 'ground'
'Ground' is not a specifically defined game term in 5e so we need to fall back on what might be considered the 'plain english reading' to understand what can be considered ground.
In the accepted, and most upvoted answer to a slightly related question What is “the ground”?, Vladislav Martin says:
'ground refers broadly to any solid continuous, contiguous surface on the planet.'
So, while it's possible that a sympathetic DM might rule in your favour (certainly for Rules As Fun reasons), probably most DMs would rule, that the top of a tree could not reasonably be considered 'ground'.
Therefore, you would not be able to cast the spell from a position already halfway up, or at the top of, the tree.
...If, as you said, you 'don't move the disk around or anything, it just stays next to [your] makeshift ladder' - you might be able to achieve much the same effect just by climbing the tree, and raining fire from there - without needing the disc at all. Visibility might be a little worse - but that would be up to your DM.