sp_getapplock is pretty much the befiest and most arbitrary lock you can take. It functions more like the lock keyword does in OOO programming. Basically you name a resource, give it a scope (proc or transaction), then lock it. Pretty much nothing can bypass that lock, which is why it's solved your race conditions. It's also probably mad overkill for what you're trying to do.
The first code/architecture idea that comes to mind is to restructure this table. I'm going to assume you have high update volumes or you wouldn't be running into these violations. You could simply use a try/catch block, and have the catch block retry on a PK violation. Clumsy, but might just do the trick.
Next, you could consider altering the structure of the table which receives this stream of updates throughout the day. Make this table primary keyed off an identity column, and pretty much nothing else. Inserts will be lightning fast, so any blockage will be negligible. You can then move this data in batches into a table better suited for batch processing (as opposed to trying to batch-process in real time)
There are also a whole range of transaction isolation settings which adjust SQL's regular locking system to support different variants (whether at the batch level, or inline via query hints. I'd read up on those, but you might consider looking at Serialized isolation. Various settings will enforce different runtime rules to fit your needs.
Also be sure to check your transactions. You probably want to be locking the hell out of this table (and potentially during some other action) but once that need is gone, so should the lock.
