I disagree with the above. Reducing the pipe size will increase the pressure loss, not the pressure, and eat up some of the
energy you want for your turbine in friction. Your starting pressure is 29.4ft of head and you can't make that go up.
At the outset I have no hydro power
experience, but I do know water and pressure quite well.
I also disagree with your dealer. Over short distances 110mm pipe may be fine (say 100ft), but your run of penstock is fairly long for that diameter pipe at those flow rates. I think you are right. My working is in metric as that is what I know, apologies.
29.4 ft is 8.96 metres head.
825 ft is 251 metre run of penstock.
236 gpm is 14.9 L/s (lower value given)
In Australia we have two types of the black poly pipe (I assume this is what you are talking about) - Rural grade (called green line) which is 800kPa (~80 metre) maximum pressure and the internal diameter is the same as the nominal diameter in this case 110mm. The metric stuff has a wider range of pressure ratings but the internal diameter changes depending on the pressure rating. It may depend on what you have available. PN 2.5 is the lightest and has an internal diameter of 104.6mm with a max pressure of 2.5 bar, about 25m. You could probably get away with that with a safety factor, or PN4 for a bit more. PN4 has an internal diameter of 101.6mm. The higher the pressure rating, the thicker the wall, the longer it will last.
I have a little calculator here. PN2.5 black poly pipe at 14.9 L/s is going to use up 5.74 metres of head in friction, more when joiners and such and such are taken into account. You will have about 8.96-5.74 = 3.2m head left, or ~10.5ft. You will have to work your turbine power out with that. this is the lower value you give. You could use larger pipe, perhaps PVC to lose less head. Plastic pipe like black poly or PVC of 150mm internal diameter (160mm metric poly) would only lose about 1m, a bit over 10% of the pressure at the lower flow rate. Bigger is even less.
Or you could find a small turbine that needed less flow. A 3kW turbine (assuming your dealer isn't exaggerating a little) makes 72 kWh a day, that is quite a bit depending on what you are using it for. 72 kWh would last me nearly two weeks of electricity consumption. Heating will chew up a bit though. Hope that helps.