Relationship Performance: Mapping based on String or Number?

LeoST it's not relevant for the relationship itself becasue the columns in the model are anyway stored with indexes that replace the original values in the column (hash encoding) and the relationship is based on those indexes. The original columns values are stored in dictionary tables that maps to the indexes. There are also more techniqes the vertipac engine use to compress the data like run lengh encoding and value encoding. The latter is not relevant for string columns. 
However, your situation applies that this big string key column also exist in both the dim and fact, which means this is a very heavy column that takes a lot of memory. It will anyway better to replace this column with a number to reduce the size of the model, which will of course help in dodging memory limit sceanrios and could help also the performance in general.
To coclude, there are probably other major reasons you have performanc issues like how you created your model and measures. It's a journey to find them all, and you of course need to start fixing the ones that cost the major impact and move to the others. This is a discipline of its own.