I know what you mean! I was horrified when I started researching exercise equipment and saw the weight limits. They really need to start building more sturdy equipment as a general rule, not just for over-priced exceptions. As you say, it's a bit of a Catch-22 when you need to lose weight in order to exercise so you can lose weight!
I'm not sure if it's safe for you to use the bike if you are 20% over its weight limit. Can you talk to the people who sold it to you? They may know if the weight limit is too conservative or not. If they didn't warn you about the weight limit you may have good cause to demand a refund.
If you choose to get another bike can't find anything for your weight in your price range there are always pedal exercisers. That way you can sit in a normal chair and still get exercise.
I have the lightest (and cheapest) of the three. The biggest problem with it is that you need to prop it up against something otherwise when you try to pedal it keeps getting away from you! I assume the other ones are far heavier to prevent that problem, but they are also more expensive at over $200 compared to under $50. However they're still much less than what you paid for your current exercise bike.