The problem with the approach of the O.P. is that it was reactive rather than proactive and as such played a role in the client opting for a lower priced option. Folks need to accept the fact that undercutting is part of a competitive marketplace--low cost alternatives exist in almost every, if not all, industries. Cottage laws which create the possibility of lower overhead and lower barriers to entry, among other factors, means the specialty cake market will have its share of low cost alternatives and thus undercutting will happen. Note costumerczar is right--undercharging and undercuttering are not the same thing--the effect of undercharging will mean your prices are undercut, but undercutting as a competitive strategy stems from knowledge of competitor's prices.
Returning to the point--given this situation it is incumbent on businesses that cannot afford to slash their prices to nonetheless develop strategies that account for low-cost alternatives. Undercutting is going to happen--the specialty cake market is not immune to competitive forces and taking the moral high ground (especially if your business has benefited from low-cost alternatives like buying supplies from an internet store that has no corresponding brick and mortar shop) is simply wishing for something that is not going to happen -- being immune to low-cost alternatives.
At least part of the problem in this situation is that the client was not made to feel like they were getting enough value (quality relative price). The OP highlights this fact by stating that she felt motivated to "educate" the client as she felt the client might have thought she was being price gouged. And indeed explanation of the client's response seems to confirm this.
The problem is the O.P. should have educated the client before-hand. This is a common recommendation for how to compete against low-cost alternatives. Given what is described here, the client did not have a sense of the quality provided by the O.P. and frankly the email does not provide much either resting the justification for prices on the fact that she is trying to make a living from her business.
Given all of the discussion of common assumptions held by the general public (it's just flour and water) and the impact of cake shows, it is poor business practice to assume that potential clients are knowledgeable about the industry generally and what sets you apart from other non-low cost specialty bakers. If you are not educating the client as a way of building value (and along with it trust) you are facilitating conditions where customers will make decisions solely on price.
Finally, the O.P. should have discussed her brand and not engaged in a sweeping generalized smear campaign (this is why there was little educational value to the email and why it thus unprofessional). Now maybe this was done at the initial meeting, if so, it was not effective as the O.P. states she felt the client assumed she was being gouged. But the "information" provided is not based on the particularities of the unknown competitors and seems to be predicated on the idea that the only legitimate business is one where which enables the owner to make a living. Not much educational value here. Rather than besmirching the competition with generalizations and assumptions that may not account for the low-cost competitor, the O.P. should have been focused on developing trust with the client by emphasizing what sets her apart and how she generates provides value to her clients.
Would this have made a difference? In this case it seems not as it seems the client's financial situation influenced her decision. Then again she make have been more willing to make other sacrifices. Point is it is poor business practice to act as if the specialty cake business should be immune to the pressures of low-cost alternatives. And if you only become interested in educating clients after they have opted for low-cost alternatives, you will continue to lose clients to low-cost alternatives.