I don't have time to go back and read exactly the definitions, but I suspect the issue here is that Brydges explicitly writes the dependence on the supporting sets, but not on the dependence fields. I recommend that review the definitions by adding/restoring in your notation this dependence in the fields, with care: namely don't blur the difference between $\sum_{k>j}\xi_k$ and $\sum_{k\ge j}\xi_k$ for example. I know it sounds silly, but if you do this carefully, you should see that some filed remains which has not been integrated over. So the result $(F')^{\Lambda}$ should be a function of that remaining field. The latter could a priori be a function of blocks of some scale (coarse field), but you can see it as a function of unit blocks (fine field) in a trivial way: the value of the fine field for unit box is the value of the coarse field on the big block that contains the unit box.

This is just a quick generic answer.

I don't have time to go back and read exactly the definitions, but I suspect the issue here is that Brydges explicitly writes the dependence on the supporting sets, but not the dependence on the fields. I recommend that you review the definitions by adding/restoring in your notation this dependence in the fields, with care: namely don't blur the difference between $\sum_{k>j}\xi_k$ and $\sum_{k\ge j}\xi_k$ for example. I know it sounds silly, but if you do this carefully, you should see that some field remains which has not been integrated over. So the result $(F')^{\Lambda}$ should be a function of that remaining field. The latter could a priori be a function of blocks of some scale (coarse field), but you can see it as a function of unit blocks (fine field) in a trivial way: the value of the fine field for unit box is the value of the coarse field on the big block that contains the unit box.

This is just a quick generic answer.