To use use advantages of inverted index, I'd suggest you full-text search engine for your purposes, e.g. Lucene or Solr (which is based on Lucene)
You can construct "document" (in terms of Lucene), which would contain fields, which associated with MinHashes of your documents (db records). Note, that you're able to index numeric fields as well (you're just need to describe field types in scheme). Also, you have to store primary key of each document, to map Lucene "documents" on records from your db.
Index entire collection of your documents.
For finding similar documents to given document - you're have to calculate MinHashes for each field, and query Lucene for similar documents:
field1:MinHash1 OR field2:MinHash2 OR ...
As more fields matched document - the higher rank it would have. So, you can take few documents with highest rank, and make a decision - if they are really similar in your case
Also, boosting of fields may be useful for you