The most appropriate statement would be:
Please activate my trial. If I am happy with it, I will switch to the paid version.
There are two separate complete thoughts (activating a trial and switching to a paid version), so there are two sentences. A conjunction would have to be used to combine them:
Please activate my trial, and if I am happy with it, I will switch to the paid version.
The word "if" begins a conditional clause, and conditional clauses can be tricky things. In this situation, you are talking about a situation that has not yet happened, but is likely/possible. Therefore, the conditional clause requires a present tense verb, followed by a future tense verb in the sentence.
Of the versions you proposed, only "am" and "become" are present tense verbs. However, the present tense of "become" is not used very often and can be considered overly formal to the point of awkwardness. If you really wanted to use it, you could say:
Once I become happy with it, I will switch to the paid version.
Here's a link with more information on conditional clauses