I found a very "hacky" way around this issue. It allowed me to add the new columns, but it ended up creating a new (minor) problem.
What I did was this: I went into the Partition settings and clicked a little "SQL" button on the right-hand side that changed the source for the table into a query. The query auto-populated as, "select star from view." I changed this query into, "select top 200 star from view." Now, this new query rendered in less than 30 seconds, and i was able to get the new columns added. Once the columns were added, saved, deployed, and safely tucked away in source control, I changed the query back to "select star from view." Voila!
While this "solved" my immediate issue, it still feels hacky. What if, rather than a lot of rows, my query took more than 30 seconds because of a long calculation or an intense group and sort? This hack would not have worked! Further, now I'm left with a query for the source of my table, and there is no way to change it back.
Again, every timeout setting I can find =0, which indicates no timeout. I know for a fact there is no timeout killing my query on the database side because I run longer queries all the time, including some that can take 10+ minutes during deployment.
What in the world is wrong with this silly data preview in Tabular? Can we kill this, or make it optional in a future release? It's awful.