A Peltier cell will provide somewhat more heat than a simple electric heater. How much more depends on how big the temperature difference is between the water and the source of heat.
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That shows what you can expect. If the temperature difference is small, less than 20 °C difference, so above around 0 °F, and if you run the Peltier cells at low currents, you can probably get 2 - 3 times as much heat out as you would with a simple resistive heating element.
A downside is that you need a heat source, as described in post #2. At those temperatures, you run the risk of the heat source icing up. You might need to reverse the Peltier cell briefly every so often to warm the heat source above freezing to get any ice that has built up to fall off. Also the efficiency will reduce quite quickly as the outside temperature decreases.
The efficiency of a solar cell is very poor, and the coefficient of performance of a Peltier cell is not very good, so you will get far less heat than if you heated water directly from a solar panel, and stored the heat in a well insulated hot tank, something like this:-
https://www.apricus.com/html/solar_collector.htm#.VybiJiMrL-k
You can store far more energy in a water tank than in a lead-acid battery. A lead-acid battery is around 40 Wh/kg. Water at 70 °C has around 80 Wh/kg of heat compared to 0 °C, and a water tank that holds 250 kg of water is much cheaper than 250 kg of batteries.
If you store hot water you would need some thermostat and probably a pump to feed hot water to the tank that the goats drink from to keep their water at the right temperature without loosing all your heat quickly.
Whatever system you use, insulating what parts of the water tank that you can will reduce the heat loss that you need to make up.