@MarcZ, just a remark: you will need two CPE 510 for a directional link, one at the top of the hill and one at your house.

A smartphone will not be able to send anything back over a distance of 250 yards. I suggest to use a wireless LTE/UMTS modem connected by wire to the CPE 510 on top of the hill. The two CPEs will cover the »last mile«. You need a power supply for both devices, the CPE 510 and the LTE/UMTS modem, maybe a solar panel which can deliver required power.

MarcZ wrote

1. Would the CPE 510 be my best choice for this scenario?

In my opinion, CPEs are the best choice for up to 100 Mbps data goodput. If there are no neighboring WiFi APs around, you can either use a pair of CPE510 (5 GHz band) or a pair of CPE210 (2.4 GHz band). I prefer CPE510 because in my country a) the 2.4 GHz band is often overcrowded and b) the 5 GHz band allows for higher signal power.

You can read this story (just skip the tech speak) to see how we get WiFi to an event location down the hill with maximum data throughput CPEs are capable of. To give you an idea: if we replace CPE510 by CPE210 on this link described there, throughput goes down by 90% due to ~140 neighboring APs in the 2.4 GHz band which are competing with the CPE210 for AirTime, thus causing interferences.

2. LTE/UMTS modem seems ideal, but will cost an extra $30/month on my plan.  Can I just use my cell phone instead?

The cell phone has no Ethernet port, has it?

I would buy a weatherproof LTE outdoor modem and put the cell phone's SIM card into this modem. Unfortunately, TP-Link doesn't offer such an outdoor LTE modem as far as I know (Penry.huang, please correct me if I'm wrong here). An outdoor LTE modem looks like this:

This LTE modem can be mounted on a pole and connects to a CPE510 or CPE210 using an Ethernet cable. The LTE modem uses Power-over-Ethernet just like the CPEs do and can be powered with a power source providing 12 to 57 volts (note that the CPEs require 24 volts). Wireless LTE throughput is up to 150/50 Mbps (might depend on your ISP plan and availability).

@MarcZ, now you know more than I do. :-)

I didn't know that there are open-all modems for multiple cell towers, but this sounds good in case a tower has problems/is overloaded. Is there a big price difference between the two models?