Question concerning an "unusual" research design

Hey everyone,
I have a question concerning a between group research-design and I wonder if it is very wise since I never came across a similar design (the provided example is of course fictional).

Basically, i want to randomly assign participants into 4 groups (one control condition). For example: Version 1, 2, 3 and 4 of a game. Each of these four groups would receive the first measurement (dv1), e.g. a questionary examining the gaming experience.

Afterwards, half of each group would receive would receive one of two versions of the second measurement (dV2). But this is where its unusual: One half of each group would receive dV2a (examining time needed to solve a math problem after playing game 1, 2, 3, and 4) whereas the other half would receive dV2b (examining attention span solving a puzzle after playing game 1, 2, 3 and 4).

I do not want to compare the dV2a and dV2b directly or make interpretation but rather see what the effects are on both measurements.

There are the following alternatives:
The first alternative could be a within-design, namely that each participant, after dV1 receives first dV2a and than dV2b. But in my case, this can't be an option since the answers to dV2a will compromise dV2b and vise versa, if dV2b was followed by dV2a.

Of course i could design two different studies (alternative 2), first study dV1 followed by dV2a and a second study with dV1 followed by dV2b. But since I am interested in the answers of dV1 of all participants I do not want to split it into two different studies.

I researched if there was a similar design, yet I haven't found an example within (mostly) social psychology research. I have thought about this design a lot, the pros and the cons, yet, even though I can't remember reading about such a design, I do not see why this shouldn't be done, especially since there won't be a direct comparison of dV2a and dV2b.

I would be glad if you could give me your opinion why this should or shouldn't be done this way.

Thank you all in advance!

Best regards, mike