What do I do if someone studied the same topic before, but did so inadequately?
But there is nothing online except some initial pages of thesis (as a blog post on the student's website [...]. It has not been published in a peer-reviewed forum, and it looks like it was done in around 4 months with a small sample size for coursework thesis. Moreover, it seems to be in a different field of study.
From the available pages it looks like research has tried to touch most of the things but at a very superficial level: three pages of data analysis and one page of findings also make me question how in-depth the analysis was.

None of this indicates that the previous study was 'inadequate'!
You should definitly mention this study in your proposal, but use better arguments than 'three pages of data analysis and one page of findings' to describe its limitations!