How to get Youtube channel details using Youtube data API if channel has custom url
Simplest solution, using API only, is to just use Search:list method of YouTube Data API. From what I can tell (mind you, this is from my own research, official docs say nothing on this subject!), if you search using the custom URL component, with "channel" result type filter and "relevance" (default) sorting, first result should be what you're looking for.
So the following query gets 16 results, with the first one being the one you're looking for. Same goes for all other custom channel URLs I tested, so I think this is the most reliable way of doing this.
Edit: Well, it returns no results for smaller channels, so it's not reliable at all.