Transferring while getting a second bachelor’s degree is rare enough that many (or most) of us probably do not have any direct experience with your situation. I certainly do not have experience with this. As such I am guessing.
However, to me it looks like you are most likely in a reasonably strong position as long as you can afford to finish your second bachelor’s degree, and as long as you want to do it. “Determination” may be as important of a skill as any that you might possess right now.
Your 3.4 GPA in your first degree is not all that bad, and how you did in the past in a humanities program does not say much if anything about how you are likely to do now and next year in an engineering program. Having a full year of a 4.0 in an ABET accredited engineering program is a good start, and suggests that you can handle the academics. Being from a WICHE/WUE state might give you some affordable out of state options to consider. All of this is positive.
If you do not succeed in transferring then you can still complete your ABET accredited engineering degree where you are.
I do not see any harm in applying as a transfer student and seeing how it goes. I do not think that professors that you ask for a reference are going to take this personally. If they are good professors then they will want what is best for you (which is something that they might want to discuss).
In terms of engineering degrees, “prestige” really does not matter. Graduating from an accredited program does matter. Handling the course work does matter. Getting some internship experience can also be a plus if you can do it.
I do not think that extracurricular activities are going to matter at all at this point. If you have any experience in an engineering related work that might help, but I do not think that it is essential.
If you want to transfer to MIT or Stanford or UC Berkeley that is most likely just not going to happen. However, if you want to transfer then I think that it is worth applying to a few engineering programs and see what happens. You will want to keep your budget in mind, which might (or might not) suggest that considering WUE programs is a good idea.
Let is know if you are successful. However, if you succeed in transferring to a school that you like better then I do understand that you might want to keep the names of both your old school and your new school private. Your situation is unique enough that naming either school might tend to defeat the “confidential” part of this web site’s name.