Consider the following regression equation: $\gamma_{ib}=\beta_{b}+\alpha_{i}$. Where $\gamma_{ib}$ is matched bank-firm loan growth between $t$ and $t-1$. $\beta_{b}$ is a set of $B$ dummies (one for each bank) and $\alpha_{i}$ is a set of $F-1$ firm dummies (where one firm is dropped to avoid perfect multicollinearity). There is multibank lending such that each firm borrows from multiple banks. Important: there is no constant. There is no time dimension.
Suppose I estimate the following two regressions seperately (OLS): 1. $\gamma_{ib}=\beta_{b}+\alpha_{i}$ where firm 1 is exluded in $\alpha_{i}$ and 2. $\gamma_{ib}=\beta'_{b}+\alpha'_{i}$ where firm 2 is excluded (and firm 1 included). Of course, $\beta_{b}\neq \beta'_{b}$. If, in a next step, I demean in the estimated $\hat{\beta_{b}}$ and $\beta'_{b}$, I find that the resulting vectors of fixed effects are the same, i.e.
$\hat{\beta_{b}}-\frac{1}{B}\sum_{b}\hat{\beta_{b}}=\hat{\beta'_{b}}-\frac{1}{B}\sum_{b}\hat{\beta'_{b}}$
Is it possible to formally show the more broader result that the demeaned bank fixed effects are the same no matter which firm fixed effect is dropped?
First, it's not clear to me why you want to de-mean fixed effects.
Anyway, the reason that the $\beta$s change is because by changing the base for $\alpha$, you're changing the "constant". Recall that even with no explicit constant term, the sum of the bank fixed effects will be 1 and thus the sum of the betas will be equivalent to a constant.
What will be left unchanged is the difference between the $\beta$s.
Therefore for every $b$
$\beta_b - \frac{1}{B}\sum_i \beta_i = \frac{1}{B} \sum_i \beta_b - \frac{1}{B}\sum_i \beta_i = \frac{1}{B} \sum_i (\beta_b - \beta_i) $
and similarly
$\beta'_b - \frac{1}{B}\sum_i \beta'_i = \frac{1}{B} \sum_i (\beta'_b - \beta'_i) $
Since $\beta_b - \beta_i = \beta'_b - \beta'_i$ for all $b,i$ these two expressions must be identical.