You've got a couple of options.
1) What chemicals are you using? If you are using the Thetford AquaKem, i.e. the blue stuff, then that smell that irritates your throat when the chemicals are concentrated is coming from formaldehyde. The smell dissipates greatly when it is diluted. Trying using less and see how that works, or use a formaldehyde-free chemical -- Thetford makes one, can't remember the name for sure, but I want to say it's CampChem.
2) I personally do not smell the toilet when the lid is down. However, camping in hot climates increases volatility of the chemicals, so you're more prone to be offended in the summer vs. winter. But unfortunately, for reasons other than your current issue, the design of the Electra Magic has remained the same for 20-30 years -- there have been no changes that I have been aware of. But fortunately for you, the toilet in my 2000 TM already has a lid seal. It's just a small black seal that goes around the outer rim of the lid, side to front to side. I can't remember if it's in the back, but since I don't smell anything when the lid is down, and I can when it's up, I can only assume that it helps.
The lid seal I'm talking about is #4 here:
http://www.aaarv.net/electra_magiv_80.htm
Unfortunately, it is very difficult to get certain parts for these toilets without buying a $200 kit that includes a bunch of other parts you don't need. But there are some places that are able to sell some of these parts individually, and I have had some luck with the above vendor. For others, I've had to order parts from a vendor in Europe, where most of these toilets are sold.
If you can't get the Thetford seal, just get some u-channel seal from McMaster and make your own. Cheap fix.
3) You can get some vent fans that run on a trickle of energy. The existing fans only draw about 0.5A, and I've seen comparable ones for twice the price ($15 instead of $7) that have better bearings, fins, etc. for computers that push comparable air and draw less -- maybe 0.25A. So even if you ran the thing 12 hrs/day, that's only 3Ah/day. For a 4-day trip, that's 12Ah. Whether that's significant depends on your battery setup and typical use habits, but I know it would not be significant for me, especially since putting in the LED lights.
Dave