You need to know the strength of the original shaft before you can look at a replacement.
If you go 316 it will need to be in the as cold drawn condition to get higher strengths.
The 416 (or 410) is quenched and tempered to get higher strength.
Depending on the original properties if you need something stronger you may need to look at Aquamet (cold drawn Nitronic 50) or 17-4PH.

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P.E. Metallurgy, Plymouth Tube

The obvious question to the OP is, why change material?

It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. (Sherlock Holmes - A Scandal in Bohemia.)

Because AISI 416 is not available in the local market and replacement is required ASAP. It will take long time to procure spare from manufacturer.
AISI 316 is readily available locally.

So you will have to tell us lot more tech data to enable an intelligent answer:
Why was the original pump shaft 416?
Pump detail, size, speed, power input, shaft dianeter/s, product being pumped.
Ask the manufacturer, tell its an urgent requirement, can you substitute 316.
You have some very smart people here, but we can only react to proper data.

It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. (Sherlock Holmes - A Scandal in Bohemia.)

My first thought is that this might be a vertical turbine pump, because 416ss is common in that.

As comparison, here are the shaft hp limitation at 3600 rpm for a 1" dia shaft:
416ss -- 142 hp
316ss -- 69 hp

So you need to consider the strength if you're going to a weaker material.

I'd want to know why the original shaft deformed, also. Was it a one-time event like ingesting solids, or wear over time?

When is stainless steel not stainless steel? There are scores of grades of stainless steel, grouped into at least four main groups and even more subgroups.
It is a fundamental design principle that any replacement component must meet or exceed all of the relevant properties of the original. In the case of your pump shaft, a critical component, strength and corrosion resistance are two obvious ones; without more information we cannot know if there are not other important characteristics.
This decision should not be made on the basis of material availability.
You take it from there.

"Everyone is entitled to their own opinions, but they are not entitled to their own facts."

Note that the thermal expansion of these metals is quite different. This can affect bearing settings, bearing clearance, etc.

EdStainless and Ten Penny answered your question.

No.

Specify a shaft with equal strength and corrosion resistance or get a new pump with the same.