If there is a threshold price called $K_1$ for which the only way the stock price could fall below $K_1$ is if the company goes bankrupt over the period. Assuming that were to happen, the stock price would fall to $\$1$, show that for any strike $K<K_1$ the put price could be close to:
$e^{-r(T-t)} (K-\$1)(\text{Probability of Bankruptcy})$
Would I use the Black Scholes model to work around this problem, or is there another method to estimate this?
This problem statement needs a bit more clarity. It sounds like that the event that the company goes bankrupt is assumed to be identical to $\{S_T<K_1\}=\{S_T=1\$\}$. When $K<K_1$ then clearly $(K-S_T)^+=(K-1\$)^+.$ Because this payoff is deterministic and attained with probability of bankruptcy we clearly have for such strikes $K$ a put price of $$ e^{-r(T-t)}(K-1\$)^+\mathbb P\{\text{bankruptcy}\}\,. $$ This holds exactly and not just approximately.