Under the Rendleman-Bartter model, a closed-form formula exists for the zero-coupon bond price. However, it is very complex involving Bessel functions and complex numbers...
Deriving the formula is actually the purpose of a paper by Uri Dothan called "On the term structure of interest rates" that you can find here: https://www.sciencedirect.com/science/article/pii/0304405X7890020X
The solution is also given in "The Lognormal Interest Rate Model and Eurodollar Futures" by Michael Hogan and Keith Weintraub that you can download here: https://www.researchgate.net/publication/269111948_The_Lognormal_Interest_Rate_Model_and_Eurodollar_Futures
It gives the price of the zero coupon $P(0, T)$ conditionally on the terminal short rate value $r(t)$:
$$
\mathbb{E} \left[ e^{-\int_0^t r(u)du} | r(t) = x \right]
$$
You can integrate over $r(t)$ whose density function you know to get the (unconditional) zero-coupon value:
$$
\mathbb{E} \left[ e^{-\int_0^t r(u)du}\right]
$$
You can add to your models list Ho Lee, CIR++, Hull-White, that enable to match the market's zero curve at t = 0. Then you can make the parameters piecewise constant and calibrate on other instruments as well such as swaptions, etc.
For interest rates modelling, I would recommend Andersen and Piterbarg's excellent book: Interest Rate Modelling (Part III - Term structure models):
https://www.amazon.com/Interest-Rate-Modeling-Structure-Models/dp/0984422110/
It is generally not recommended to work with lognormal short rate models (such as Rendleman-Bartter, Black-Karasinski, Black-Derman-Toy), as they give an infinite expectation for the inverse of future zero coupon bond prices:
$$
\mathbb{E} \left[ \frac{1}{P(s, T)} | \mathcal{F}_t \right] = +\infty, t < s < T
$$
Furthermore, Rendleman-Bartter doesn't have the mean-reversion property empirically observed in interest rates.
