In the case of a moving conductor cutting magnetic flux, you can derive Lenz's law without obvious appeal to energy conservation. A simple set-up is a metal rod sliding on metal rails connected together by a resistor at one end. The rod, rails and resistor form a rectangle. A uniform magnetic field is applied at right angles to the plane of the rectangle.
Apply $\mathbf F_\parallel =q\mathbf v_\perp \times \mathbf B$ (or Fleming's left hand motor rule) to the charge carriers in the rod, whose velocity component, $\mathbf v_\perp$, at right angles to the rod is that of the rod itself as it made to slide along the rails by an external force. The force, $\mathbf F_\parallel$, on a carrier (if positively charged) acting along the rod is clearly the direction of the emf. The current driven by this emf gives rise to a force opposing the motion of the rod – we use $\mathbf F_\perp=q\mathbf v_\parallel \times \mathbf B$ (or Fleming's left hand motor rule) on the component, $\mathbf v_\parallel$, of carrier velocity along the rod. This agrees with with Lenz's law.
Algebraically we have
$$\mathbf F_\parallel =q\mathbf v_\perp \times \mathbf B$$
But the drift velocity, $\mathbf v_\parallel$ of carriers along the moving rod will be proportional to $\mathbf F_\parallel$, so we can write
$$\mathbf v_\parallel = a\mathbf F_\parallel\ \ \ \ \ \ \text{in which } a \text{ is a positive constant}$$
Therefore the force on the carriers due to their motion along the rod will be
$$\mathbf F_\perp\ =\ q\mathbf v_\parallel \times \mathbf B\ =\ qa\mathbf F_\parallel \times \mathbf B\ =\ q^2a(\mathbf v_\perp \times \mathbf B) \times \mathbf B\ =-\ q^2a\ \mathbf B \times (\mathbf v_\perp \times \mathbf B) =\ -q^2 a\ B^2 \mathbf v_\perp$$
• The last step used the BAC-CAB rule.
• The minus sign, embodying Lenz's law, arises whether the charge carriers have a positive or negative charge, as it is $q^2$ that appears.
This argument cannot, of course, be used for the other kind of e-m induction, where an emf is induced in a stationary circuit by a changing magnetic field. Here the emf arises from an induced electric field in the circuit. The electric field can be calculated using the Faraday-Maxwell equation $$\text{curl}\ \mathbf E = -\frac{d\mathbf B}{dt}.$$
The minus sign embodies Lenz's law, and the usual way to justify it is ... by appeal to energy conservation!
In certain special cases, such as thrusting a magnet into a coil, we can justify the minus sign by invoking the relativity principle...
The rod-on-rails argument above can be generalised to give the general equation for a moving circuit of any shape:
$$\mathscr E =-\frac {d\Phi}{dt}$$
This applies if we move the coil in the (non-uniform) field of a stationary magnet. According to the Relativity principle the same equation – with its minus sign – will apply if we move the magnet and keep the coil stationary. I'm unable, though, to extend the argument to include, for example, mutual induction.
