That's what everyone said about installing a block off plate too
Except I think given an external chimney, where it is clear the liner is going to lose heat, the way to prioritize would be:
1) insulate liner
2) add block-off
Given your dilemma, this in hindsight obviously makes even more sense with a basement install with a greater likelihood of downdraft. It's good of you to post your observation, because I think you are correct, this scenario of bad draft performance due to a block-off is not ever mentioned. But bholler is right, by not having a block-off and insulation you are basically paying to heat the great outdoors 24/7 just so your chimney can be warm enough to draft.
When you said that you were running only one fire a day, was that normal or just during that month because you had to preheat the stove? Because if you don't burn full time, and always have some cold-stove time with no coals, that is what allows the total cooling and reversal of draft. I would think even just a few coals hours after a productive fire might provide enough warmth to keep the draft, so if you were burning more hot and more often, if not full time, you might not experienced the same bad block-off effect.
An inexpensive way to insulate your existing liner would be with loose perlite, but be aware that this is likely not an approved insulation method per the liner manufacturer, and thus not code compliant. It sounds like you are not using the stove to heat your entire house, so probably not a big deal, but I suspect that running the stove is not saving you as much on your heating bill as it would be with both insulation and block-off.