The first part of the answer, why the setup is not stable, is here: [RBC with Government Sector).
The second part is: the timing difference comes from the fact that nominal bonds are risk-free for agents. Bonds holdings decided today, b, were promised to return tomorrow a nominal interest rate R, that was decided already today. This makes them nominally risk-free in the Euler equation. But the inflation part makes them risky in real terms.
This differs from capital decided upon today, k. It will return a stochastic return whose realization is only known tomorrow, r(+1).
In what you wrote down, the government budget constraint is wrong. It should be
See e.g. equation 22 of Born/Peter/Pfeifer (2013): Fiscal News and Macroeconomic Volatility, sciencedirect.com/science/ar … 8913001437