I'm assuming fresh gas, gas filter is new, air filter is cleaned, gas tank is properly vented and your existing carb has been cleaned many times. These are the "quick win" things to look at first.
Couple of things you can do to this 2 stroke.
1 - Get its engine PSI tested. With the PSI compression numbers, ask around if its engine is running within its normal operating range. Sorry, I don't know the PSI for your "beefed up" engine. But for a pure stock '03 Pred-90, normal operating range is 90 to 110 PSI. 120 is fresh and anything below 90 is loss of power. Anything below 65 is a hit/miss if it will run at all. If wondering, PSI tests the compression in the top of the pison area. This includes head leaks, worn rings and/or worn out cylinder walls. For the PSI tool I use, surf:
Note: The place you bought the "beefed up" cylinder / head should tell you what its normal PSI should be.
2 - Perform an engine compression test. Being a 2 stroke, its internal engine must have internal pressure. The normal pressure is 6 lbs. If you have a gasket leak on intake and/or leak on its engine seals, this test will most often find it. For more details on this test, surf:
Note: If you rebuilt your own engine, I'd perform a pressure test on it. When a shop performs an engine rebuild, they do perform this pressure test (well, they are suppose to).
3 - If you confirm your engine is within PSI "normal range" for your beefed up engine and there's no engine air leaks, the problem is most likey with its carb. Here's a few things to consider / investigate:
A - After you installed the upgraded air filter, did you re-jet the carb? After replacing the factory air box, you must rejet its carb to obtain a "light black to dark brown" color in all throttle ranges. For more details on spark plug colors, surf bottom page of:
4 - Depending on the factory size of its carb, you may have to drill out the main jet hole and/or float bowl hole when going above a certain Jet size. Sorry, I don't know the jetting details for your carb. But I do know that on my '03 Pred-90 16mm carb, I must "drill out" the main jet hole if upgrading above 90 Main jet. And if it feels like its running out of gas above 3/4 throttle, I have to drill out its Float jet hole. To allow ease of jetting on my '03 Pred-90, many carb pros have told me to upgrade its factory 16mm carb to 20mm carb. The 20mm carb is much easier to jet and maintain then its factory 16mm carb. You may have to upgrade the carb on your mini as well.
5 - With any 2 stroke engine, carb jetting is dramatically influenced by altitude changes and weather temp changes. Much more then the same carb on a 4 stroke engine. Therefore, your mini may run great one day then fall flat on its face the next day. Depending on the weather temps, its sometimes a hit / miss if it will run at all - even though you had its carb jets set perfect last week. The smaller size carbs are influenced "much more" compared to a larger size carb. Thus, a 20mm carb has a wider range of "influence" then a much smaller mm carb. In the end, you may want to upgrade the size of your carb. If I had the spare dollars, I'd replace the 16mm carb on our Pred-90 with a 20mm carb without hestitation. Something to think of... BTW: For your beefed up engine and smaller CC, the 18-20mm carb size sounds great to me. 18mm (same as the factory E-Ton 90cc) is probably the most optiminal size.
Note: Problem might also be with its stator and/or ignition coil. Or some other electrical part. But most often when these electrical parts blow, they are in an off/on condition. In other words, do or don't work.
Hope this helps....