Eradicating poison ivy can be a real challenge, especially for people who are allergic or highly sensitive. I did some research and was unable to find anything definitive about the degree to which composting would break down the oily toxin in poison ivy (urushiol). Some are of the opinion, and I agree, that it would be more likely to break down completely in a large-scale commercial composting operation than in a home compost pile. Most home compost piles do not achieve the consistently high temperatures that commercial operations do, nor do they get same degree of turning, mixing, aeration, and so on, that accelerate decomposition. Do you know whether your county or region has a commercial composting facility?
 
Because the oil is easily spread, I would not recommend mowing or shredding it. You probably already know this, but you should definitely not burn it. Even if you are not actually touching the plant when you are mowing it, the process of mowing it may actually release more of the toxin, which will get on parts of your mower including the bag. Since you are allergic, this could present problems.

Your idea is good, and there aren't many materials that I would discourage from composting, but this is one. I just wouldn't risk it.

Here is an article from the University of Arkansas Cooperative Extension Service that provides some guidance on eradication. You should feel free to contact your local Extension office for advice or clarification. Needless to say, you should plan your attack carefully and wear protective clothing or gear if you are going to do it yourself!

http://www.arhomeandgarden.org/plantoftheweek/articles/Poison_Ivy.htm