Dig that trench a whole bunch deeper and wider and go in both direction, perferated pipe and fill the trench with crushed gravel.
Just some rough calculations here if the area under roof and the slab was 20x30 or 600 sq. ft. and you got a rain fall of 1 per hour that would be approximately 400 gallons of water per hour. With a 50 gallon drum that would need dosed 8 times per hour or approximately every 8 minutes. or a 6 gallon per minute flow rate pump running continually during that rain. The idea behind a larger dosing tank and higher flow pumps is based around duty cycle. Some pumps are not rated to run nonstop for long periods of time. The estimate of one inch of rain per hour is also not any kind of record rain fall for Florida there are reports of 3 to 6 inches per hour during storms with flooding conditions.
These type calculations are why you see a 5 acre retention 10 foot deep pond next to a Walmart now. Where land is a premium the ponds are going in under the parking lot. We have a new dairy queen here and the whole parking lot is gravel except a tiny road for the drive thru. I asked them when they would pave the lot and they said never. The cost was unbelievable as they would have to buy several homes and tear them down to build a retention area for rain water.
If the OP was to try and leach the water away by digging wider and deeper trenches on that narrow strip of lawn he may well not have enough area. You also have to take into account that it is raining on that area at the same time. Just like a septic system I would have the soil tested for peculation times and then decide if that plan was feasible.