The use if the LT5400 for the gain is an interesting solution. Combining a separate resistor with the LT5400 array is not ideal, though may still be acceptable for only some 2% of the final voltage.
The suggested LT1113 in the plan is however not a good choice, as it has too much 1/f noise. I would more look at LT1013, OP07, OPA202, OPA207 or ADA4077 or maybe an auto zero type like LTC2057 or MCP6V51.
It would make sense to plan for at least some filtering and an output stage that is capacitive load tolerant like discussed before.