If being online is an issue i.e availability 99.999% or similar. Most hardware vendors will provide machines on a short term lease arrangement. The added benefit is that most data and rebuilding/testing can then happen in parallel.

If you have a spare machine, go ahead and normalise the application tables, and then run some benchmark queries, record the results. If you have time, also run some queries which would be impossible in the current database.
This will show your boss:
1) You are proactive in tackling issues
2) Hard data on timings = time saved = $$$ saved.
3) The extra information available by being in the new structure.

The problem with orginal table structure can be solved either by views or materialised views depending on your Oracle version i.e. the application will never know the difference.
This also gives more time to recode the app if required.

Have Fun