I have a theoretical question concerning usage of the FE model for panel data regression.

I'm currently dealing with panel data from 2003 to 2012 where I'm trying to analyse the effect of the borrowers gender on certain dependent variables like Operating Expense Ratio, Operational Self Sufficiency etc.
I have set the different MFIs I'm dealing with as the group variable with xtset and I have included dummies for the years with i.fiscalyear (where fiscalyears is my variable for the years from 2003 to 2012). Doing a hausman test in Stata advises to use FE as RE is inconsistent under H0.
Now I have read up on a lot of literature concerning FE vs RE, however my theoretical knowledge is not broad enough to understand the following problem:

I would like to analyse if and how the borrowers gender affects the above named dependent variables. However I'm wondering, as most of the MFIs have a relatively consistent amount of women borrowers (e.g Grameen Bank exclusively deals with women borrowers, other MFIs also have a set percentage of women borrowers, and the remaining only have small differences between the years) would FE regression really give me the results I'm looking for?
Because as far as my knowledge goes, FE regression only 'measures' within group effects.
Now my thought process is as following, as the amount of women borrowers stays relatively or sometimes even exclusively the same over the years, the 'within' effect is not what I'm searching for, as I would like to measure the 'whole' effect of the amount of women borrowers, and analysing the 'within' effect would basically only analyse the effect of the years (I'm having a bit of a problem with formulating my thought process here). So would using the RE model be better as it also measures the 'between effect' even though hausman advises otherwise?

Now I may be complete wrong here or misunderstood what 'withing group effect' and 'between group effect' means, but most of the forum posts deal with different problems concerning RE vs FE so I'm at a bit of a loss here.
Thank you in advance!