Optimisation using value function
Starting with your original equation:

$max_{c_t, m_t, b_t} E_0\sum_{t=0}^\infty U(c_t, m_t)$

s.t.

(1) $y+\frac{m_{t-1}}{1+\pi_t}+\frac{1+i_{t-1}}{1+\pi_t}b_{t-1}=c_t+m_t+b_t+\tau_t$

Here: $R_{t-1} =1+i_{t-1}$ and $1+\pi_t=\frac{P_t}{P_{t-1}}$

Note that in this problem, you have have two state variables, $m_{t-1}$ and $b_{t-1}$, and your main issue have been that you have bunched these together. your Bellman should be:

$V(m_{t-1},b_{t-1})=max_{c_t, m_t, b_t} U(c_t,m_t)+E_t\beta V(m_t,b_t)$

s.t. (1)

You can here use your contstraint to get rid of one control, or you can solve it using the lagrangian. If you use your constraint to substitute for $c_t$, your updated problem becomes:

$V(m_{t-1},b_{t-1})=max_{m_t, b_t} U(c_t(y, m_{t-1}, b_{t-1},m_t, b_t, \tau _t),m_t)+E_t\beta V(m_t,b_t)$

For clarification:

$c_t(y, m_{t-1}, b_{t-1},m_t, b_t, \tau _t)=y+\frac{m_{t-1}}{1+\pi_t}+\frac{1+i_{t-1}}{1+\pi_t}b_{t-1}-m_t-b_t-\tau _t$

FOCs:

$[b_t$]: $-U_{c_t}+\beta E_tV_{b_t}=0 $

$[m_t]$: $-U_{c_t}+U_{m_t}+\beta E_tV_{m_t}=0$

Envelopes:

$[b_{t-1}]$: $U_{c_t}\frac{1+i_{t-1}}{1+\pi _t}\implies V_{b_t}=U_{c_{t+1}}\frac{1+i_t}{1+\pi _{t+1}}$

$[m_{t-1}]$: $U_{c_t}\frac{1}{1+\pi _t}\implies V_{m_t}=U_{c_{t+1}}\frac{1}{1+\pi _{t+1}}$

You should be able to combine these equations such that you find the demand for money. To me, your equilibrium condition (along with the fact that no capital is present in the set-up) impies that there is no saving in real variables, and hence you have nothing that can pin down the real interest rate, and you therefore lack information to find the fisher equation.