A light sport aircraft pilot reported a rough running engine on initial climb and a return to the airport. While landing they had a brief runway excursion.

Narrative

I had a loss of power in flight today in Aircraft X. Here is the best of my recollection. Preflight; taxi; and run up were all normal. I elected to take-off on XXL at ZZZ due to a long line of aircraft waiting in line for XXR. Initial application of power and take-off roll were all normal. At approximately 350 ft. so about 250 AGL the engine began to run roughly; power cycling up and down a bit with a rhythmic noise. In the past; when the engine has 'skipped' a little; brief adjustments to power have resolved this; so that's what I tried this time; briefly pulling the throttle back and putting it back in. This time; though; power reduced to about 2500 RPM and refused to come back up despite application of full throttle. Both my passengers reported hearing a 'click' sound when I adjusted power; but I don't remember hearing it.It took me 3-4 seconds to assess the situation; during which we continued to climb a bit up to about 400 ft. and during which airspeed decayed to about 66-68 kts indicated. I pushed the nose down to maintain airspeed and [advised ATC]. Given my low altitude; I ruled out using the Ballistic Recovery System (BRS); as I knew it would be ineffective. Taking off on [Runway] XX at ZZZ; there is no safe landing zone straight ahead; and since this is my home field; I was well aware of this fact. I briefly considered a right turn to land on [Runway] YYL; but since I knew there were other aircraft taking off from XXR; I ruled this out. So I immediately began a left 180 degree turn in an attempt to land on YYR.I was able to get the airplane over the approach end of YYR at approximately 75 to 100 ft. but I was not lined up with the runway at this point. I was at an approximate heading of 080. I corrected to the right and was able to get over the runway with just a few feet of altitude. I was landing with a tailwind; and with only one notch of flaps; I had never retracted flaps since take-off; so I landed quite long; with less than 1000 ft. of runway remaining. Immediately on touchdown; I applied heavy braking force. Nevertheless; I departed the runway into the grass by a few feet approximately one plane length. At this point; the engine was still making 2500 RPM; which was enough power to taxi back onto the pavement and back to my hangar. Shutdown was normal. At no time did any EFIS errors indicating an engine problem come up. Loss of power was not accompanied by any warnings either before or during the event. There was no damage to the airplane; and no one was injured.