Deriving intertemporal budget constraint from flow constraint
Take your second equation, move it forward one period, and rearrange. You get:

$$ B_t = \frac{p_{t+1} s_{t+1} + \Delta M_{t+1}}{R_{t+1}}  + \frac{B_{t+1}}{R_{t+1}} $$

Then, define the nominal primary deficit as $D_{t+1} = -(p_{t+1} s_{t+1} + \Delta M_{t+1})$. The above transforms into:

$$ B_t = -\frac{D_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$

which has the format you are looking for. To find the infinite-period intertemporal budget constraint, simply proceed with forward substitution.

The notation I've used (regarding the deficit) is taken from Wickens, Chapter 5. The above is roughly equal to equation (5.11) in the book. There, the author has a more general expression, which includes inflation, GDP growth, and debt-to-GDP ratio. Your example is a simplification of his. I suggest you go through Section 5.4 of the book, which it gives a very detailed analysis of the Government budget constraint (it also gives the solution you are looking for, in case you want to compare).



To find the final expression you are looking for (after edit), take your second equation and replace $B_t$ with the infinite-period intertemporal budget constraint:

$$\frac{R_{t}B_{t-1} + M_{t-1}}{p_{t}} = \frac{B^*_{t} + M_{t}}{p_{t}} + s_{t}$$ 

where $B^*_t$ is the result of the forward substitution exercise.