It is hard to give precise necessary and sufficient conditions for the spectrum of the TISE to be bounded from below, but here are some important cases that are definitely unbounded from below (and hence excluded from OP's sought-for list):

1. If the potential $V$ has an attractive singular pole of order more than 2, i.e. if $$V({\bf r}) ~\propto~ - |{\bf r}- {\bf r}_0|^{-\alpha}\quad \text{for}\quad {\bf r}~\to~ {\bf r}_0$$ and $\alpha>2$, cf. e.g. eq. (4) in my Phys.SE answer here.

2. If the preimage of the potential $V$ has the following property $$\exists \epsilon >0:~~ \lim_{E\to -\infty}{\rm Vol}( V^{-1}(]-\infty,E]))~\geq~ \epsilon .$$

OP's linear perturbation example is prohibit by the 2nd case.

It is hard to give precise necessary and sufficient conditions for the spectrum of the TISE to be bounded from below, but here are some important cases that are definitely unbounded from below, i.e. it has no ground state and is unstable (and hence excluded from OP's sought-for list):

1. If the potential $V$ has an attractive singular pole of order more than 2, i.e. if $$V({\bf r}) ~\propto~ - |{\bf r}- {\bf r}_0|^{-\alpha}\quad \text{for}\quad {\bf r}~\to~ {\bf r}_0$$ and $\alpha>2$, cf. e.g. eq. (4) in my Phys.SE answer here.

2. If the preimage of the potential $V$ has the following property $$\exists \epsilon >0~\forall E\in\mathbb{R} ~\exists {\bf r}_0:~~ V^{-1}(]-\infty,E])~\supseteq~ {\rm Ball}({\bf r}_0,\epsilon) .$$

OP's linear perturbation example is prohibit by the 2nd case.