Constant SMC resets required
It is good you mentioned the Check Battery Warning since that is your problem.

Mac will cut down the CPU speed to 1/2 if the battery is not up to snuff.

Your SMC reset will ratify the issue for a while, but since the battery is on the verge, it will eventually trigger the cutting the CPU Speed/Power consumption to 1/2 of normal. You see that as computer slowdown.

If you want to verify my claim there is a CPU test made by Intel to check the CPU and do that during the slow times as comparison to when it is fast.

Download it, and yes you can keep it :)

Here is a sample what it looks like on my MBA.