Ranking rows with non linear order

I have to rank these rows:

In this particular way:

Without the 'Cat' type, it's a normal RANK() SQL Function.

The 'Cat' in the ranking should not stole a position of any other dogs. 'Cat' type must have the same rank of the next 'Dog' type regardless of the score.

The RANK() functionality is required for all 'dog' type (in the example Buddy and Duke are 4th and Max 6th)

A possibility is create a temporary table/stored procedure and post-elaborate results... but I would like to know if there is an easier way to to this.

Adjust the Cat score to be the following Dog score, then rank. (Or dense rank... your desired results don't match either as it stands).

WITH cte AS (
    SELECT
        Type,
        Name,
        Score,
        MAX(CASE WHEN Type = 'Dog' THEN Score END) OVER (ORDER BY Score ASC) NewScore
    FROM YourTable
)
SELECT Type, Name, Score,
  DENSE_RANK() OVER (ORDER BY NewScore DESC) DenseRank,
  RANK() OVER (ORDER BY NewScore DESC) Rank
FROM cte
ORDER BY Score DESC;

Returns:

Oh I may have misunderstood... from your desired results, maybe you want to rank the Dogs first, then the Cats just join in... in that case this is what you need

WITH cte AS (
    SELECT
        Type,
        Name,
        Score,
        CASE WHEN Type = 'Dog' THEN RANK() OVER (PARTITION BY CASE WHEN Type = 'Dog' THEN 1 END ORDER BY Score DESC) END DogRank
    FROM YourTable
)
SELECT Type, Name, Score,
    MIN(DogRank) OVER (ORDER BY Score ASC) Rank
FROM cte
ORDER BY Score DESC;

Returns:

DBFiddle