There are important tips you need to understand when using invisible thread. I think you should give that another try, especially since any mistakes you make will not be noticeable.
First off, which brand and type of invisible thread did you try? My personal favorite is YLI nylon monofilament because it is the most invisible. However, many people prefer a polyester invisible thread (such as Superior's mono-poly). Many off-brands of invisible thread do not work well for invisible machine applique.
Both nylon and polyester invisible threads stretch more than regular cotton thread. Because of this, how you wind your bobbin and how you set your machine tension are very important. When winding a bobbin you want to wind very slowly. This helps prevent the thread from stretching as it is loaded onto the bobbin. You also want to wind the bobbin only 3/4ths full, especially if your bobbin is made from plastic (metal bobbins can hold a little more, but you still want to underfill). Overfilling the bobbin (especially a plastic bobbin) with nylon or polyester can cause the bobbin to distort.
You definitely want to lower the upper tension when working with invisible thread. Lower it a *lot* if using nylon thread. Polyester doesn't stretch quite as much, so you don't have to lower the tension as much as nylon, but you still need to make the tension less than for cotton thread.
You also may need to reduce bobbin tension. How you do this depends on whether you have a drop-in type bobbin or the other kind. But I would try reducing just the top tension first.
Good brands of both nylon monofilament and polyester thread are strong and will last just fine through as many washings as cotton thread. Quality is important in invisible thread, so I recommend sticking with YLI or Superior.