What's the formal process to determine the IC by manufacturer code?
There is no formal process, unfortunately.  Manufacturers may use numbers, letters, punctuation, and arbitrary shapes as meaningful symbols in top marks, I've seen all of these.  Some manufacturers include datecodes, manufacturing location, lead-free status, etc, in addition to the part number in their top marking.

There are a few websites that provide lists of top marks with manufacturer/part number, however these are invariably incomplete.  Keeping such lists up-to-date would be a massive effort.  However, they can be worth checking, as you may get lucky.  They are especially useful with parts that use numeric markings, such as yours, since numbers are unlikely to provide useful web search results.

Speaking of web searches, that's another option.  I have had reasonable success in the past using search strings that include "IC top mark" and the marking in question.  It may help to include the package as well, although there are usually three or more different designations for a given package that the manufacturer may use, which makes that a bit more difficult.   This method works best when the manufacturer includes the top mark in their datasheets in a way that search engines can index.  Some manufacturers are better about this than others.

Some manufacturers do provide an option to look up a part by its top mark, so if you can narrow the part down to a few manufacturers, it may be worth seeing if they offer such a tool.

If you can narrow down the function of the part, you may be able to use distributors' catalogs to narrow the search based on functional information (for instance, knowing it must have a voltage rating of at least X) as well as package to some number of manufacturers and part numbers, and then it's a matter of sifting through the datasheets and seeing if any of them have the right top mark.  

The final option, is to simply not worry about what the specific part is, and focus on finding a functional replacement.  This depends on a sufficiently thorough understanding of the circuit in question, but may give you a lot more flexibility in sourcing a replacement.  You may wind up needing to go this route anyway if it turns out the part is obsolete.