How can I isolate the inputs of my mixer to prevent this?

A single-diode mixer is a "no-balance" mixer: Local oscillator, R.F. input, I.F. output appear at all three ports. A single-balance mixer allows some isolation between two of the three ports. A doubly-balance mixer gives some isolation between all three ports.

Single diode mixers have been used back when transistor amplifiers had too-little gain at VHF frequencies.
Not a reasonable design choice today.

How can I isolate the inputs of my mixer to prevent this?

A single-diode mixer is a "no-balance" mixer: Local oscillator, R.F. input, I.F. output appear at all three ports. A single-balance mixer allows some isolation between two of the three ports. A doubly-balance mixer gives some isolation between all three ports.

Single diode mixers with no balance have been used back when transistor amplifiers had too-little gain at VHF frequencies.
Not a reasonable design choice today.

^{simulate this circuit – Schematic created using CircuitLab}