I would suggest collecting performance metrics on this server, so you can eliminate the guesswork from troubleshooting these types of problems. See this article for a more complete guide if you don't know where to start with this.
In particular, I would check the performance counters
Memory\Available MBytes and
Paging File(_Total)\% Usage because you said the issues only start occurring when the buffer pool is full. The numbers you get back from these counters may indicate that the max server memory setting needs to be adjusted (either up or down) for the amount of physical memory allocated to the server. As I mentioned here, I don't recommend basing the max memory setting on the amount of physical memory except as an educated guess for a starting point. Always measure the result, and adjust from there.
If the amount of free memory is too low (< 500), or the page file usage is over zero, this may indicate that the SQL Server instance is overcommitted: on SQL Server 2008 R2, the max server memory setting only controls buffer pool size, and not other memory usage such as the plan cache. SQL Server also does not care about other applications you may have running on the system. This extra memory usage can put memory pressure on Windows -- or the other applications -- possibly leading to disk swapping. This is something you want to avoid at all cost, particularly if the page file exists on a volume backed by just a simple RAID 1 mirror. My feeling is that this is the issue, and backing off the max server memory setting should fix the problem.
If the amount of free memory is high (> 1000) and the page file usage is zero, you can probably bump up the max server memory slightly (in 256 MB increments) to maximize the memory usage of the server. This most likely won't solve the problem, however, and you'll need to look elsewhere, probably at the physical disk counters and the buffer pool page life expectancy. If queries are thrashing the buffer pool, there's nothing you can do except improve disk performance, increase the amount of physical memory available to the server so all the data pages can fit in memory at once, or modify the database to not take up as much physical space (maybe by using row or page compression, or by rebuilding indexes with a higher
I've published an article on this topic here that goes into more depth about this issue and how to solve it.