Upvote:2
The shortcomings that resulted in your failed Schengen visa application are not the same as the factors considered for residence in the UK.
As you note, your travel plans were questioned, and you, or your agent, didn't offer proof of available funds or medical insurance coverage.
The long-term right to remain in the UK is based on far greater criteria than those for a short-term visit to a country. The recent Schengen visa refusal is unlikely to impact a request to extend leave to remain in the UK.