There isn't a single reliable way that fits all circumstances. What we can do, is look at some of the factors that influence the pros and cons.
Uncertainties within the green electricity tariff
If it was absolutely clear that some of the money you spend on electricity goes to increasing the amount of renewable energy going into the grid, then that would be a big plus. However, it's rarely clear that that is the case.
Washington State US is in the WREGIS trading area, and that means that a green-energy tariff there is typically based on trading Renewable Energy Certificates on WREGIS. You'd have to do some digging to find out how much of your tariff premium actually goes to the renewable-energy generators, and how much additional capacity (if any) gets built as a result of that money. The answer there, will make a huge difference to answering your question.
And it is about how much new renewable capacity gets built, because generally, renewables have extremely low short-run marginal costs - lower than the wholesale price of electricity - which means that they generate as much as they can anyway, so increasing short-run demand for renewable electricity, does not increase its short-run supply.
Propane - useful but filthy and unsustainable
Propane summary: if your green electricity tariff doesn't get new renewables built, and the marginal electricity generator in your area is coal, then it may be better to stick with propane for now
Propane is a fossil fuel, and so we know it's a contributor to catastrophic climate change, as well as being a resource that's being depleted unsustainably. So far, so bad. But a switch away is only worth it if it will reduce overall greenhouse-gas emissions. The problem is that if your green electricity tariff results in virtually no extra money going to renewable generators, and thus no additional renewable electricity, then your additional electricity demand from resistance heating or heat pump, will actually get met by fossil generation; and the sustainability impact could be even worse than using propane.
Electrical resistance heating - very low capital impact, potentially high-impact operating impact
Resistance heating summary: If there's a net sustainability benefit from your green tariff, then electrical resistance heating offers the best contribution toward sustainability (but it will cost you money). Otherwise, it's the worst option
An electrical resistance heater and storage tank are very cheap to buy, but in operation are very intensive in terms of exergy in to energy out: electricity is a high-quality carrier of energy, whereas the low-grade heat in a hot-water tank is of lower quality - that is to say, there's less exergy per unit energy. To put it another way, getting low grade heat at an efficiency of 90-99% from electricity isn't really very efficient, despite seeming so from those numbers. On the plus side, the materials to make a resistance heater are very simple, so the whole-life sustainability impacts of the equipment, putting aside for one moment the cost of the electricity consumed, are very low.
The net sustainability impact will all depend on what the net result of your green electricity tariff really is.
Heat pump - efficient and complicated
Heat pump summary: This might be the cheapest direct financial cost to you, if the heat pump is good enough; but is unlikely to be the best option from the sustainability angle.
Relative to resistance heating, a heat pump has worse sustainability impact for the equipment up front, and probably on disposal too. Actual real-world performance could give a Seasonal Performance Factor of anywhere between 1 and 5. That's equivalent to an average annual efficiency of 100-500%. The efficiency is influenced by the temperature of your heat source (higher is better), the temperature of your incoming water (higher is better), the temperature you're heating the water to (lower is better), the quality of the installation and equipment, and how well-specified the heat pump is.
Looking forward to the Smart Grid
Domestic electricity consumption that has the ability to turn up or down in response to the grid balancer, so-called Demand-Side Response, is going to have an economic and sustainability value to the forthcoming Smart Grid. If you've got lots of storage hydro on the grid already, then that value will be low. If you haven't, and if grid-balancing at the scale of seconds to minutes is done by fossil fuel, then the value could be high. In this case, electrical heating, and in particular resistance heating, is likely to be one of the better opportunities for Demand-Side Response.
If you've got enough unshaded south-facing surface, then solar thermal could provide a big chunk of your water-heating needs, depending on local climate and your seasonal usage pattern of hot water, in a more sustainable way than any of the other options. (solar thermal works economically in Denmark, and that's at a more northerly latitude than Washington State US - but the Danes do take advantage of the economies of scale).
If you live in a high-density built environment, then getting together with your neighbours to set up a community heating scheme is probably the most sustainable option. That would allow you to get heating from a combination of different sources; and you could tune the combination to give the most sustainable outcome. The combination might feature ground-source or water-source heat pumps, solar thermal, electrical resistance heating, biomass (possibly with CHP), and maybe geothermal too.
Some assumptions I've made, in order to get to some kind of answer
- Your propane (LPG) comes from oil and/or natural gas, with the unsustainability that goes with that
- When you say you're in WA state, that means Washington State in the US, rather than Western Australia or anywhere else. I think your reference to hydro means that Washington is more likely than Western Australia.