This is a tricky question that requires careful consideration of the National Electrical Code (NEC) and potential local amendments. Here's a breakdown

Understanding the Situation
300A Breaker

This is the

feeder breaker protecting the conductors from the utility. #350 kcmil Conductors

These conductors are sized to carry at least 300 amps (based on typical ampacity tables, assuming copper and appropriate insulation).

Panel MCB (Main Circuit Breaker)

This is the main breaker within the panel you're asking about. It protects the panel's busbars and branch circuit breakers.

NEC Considerations1. NEC 220.87: Determining Existing Loads

While not directly addressing the MCB size, if the panel is replacing an existing panel, this section allows you to use actual load data to justify a smaller feeder (and potentially a smaller MCB, though not likely). This requires a specific load study performed over a defined period.2.

Panel Busbar Rating

The panel itself must have a busbar rating that equals or exceeds the calculated load. The MCB must not exceed the panel's busbar rating.3.

Feeder and Overcurrent Protection Coordination

The MCB in the panel serves two main purposes: (1) disconnect the panel from the source, and (2) provide overcurrent protection for the panel's busbars and distribution equipment.

Key Arguments and Analysis Argument for 300A MCB

The primary argument for a 300A MCB is that the #350 kcmil conductors are rated for at least 300A. If you install a 250A MCB, you are effectively limiting the capacity of those conductors and the system.

Argument for 250A MCB (Less Likely)

In most cases, you cannot reduce the MCB size to 250A. However, in the unusual event that your calculated load is significantly less than 300A and the panel's busbar rating supports it, it might be permissible, but only with very careful analysis and documentation.

Practical Considerations and Recommendations1. Calculated Load

The most crucial factor is the calculated load for the panel according to NEC Article 220. This includes all the branch circuits and their anticipated load.
2.

Panel Busbar Rating

The panel's busbar rating MUST be at least equal to the calculated load. It's typically stamped inside the panel.3.

Feeder Protection

The MCB should be sized to protect the busbars and distribution equipment within the panel. If the panel is supplying significant motor loads or loads with high inrush currents, the MCB size may need to be increased to prevent nuisance tripping.4.

Coordination with Utility

Consult with the utility company. They may have requirements regarding the main breaker size in relation to the service entrance.5.

Documentation

Thoroughly document all calculations, load studies, and any deviations from standard practice. Keep this documentation with the panel for future reference.

General Advice Err on the Side of Caution

Unless you have a very compelling reason and a very low calculated load, it's generally safer and more compliant to use a 300A MCB, matching the feeder breaker and conductor capacity. It also provides headroom for future load increases.

Consult with an Electrical Engineer

For complex situations, or if you are unsure about the correct MCB size, it's best to consult with a qualified electrical engineer.

In conclusion, while technically possible under very specific* and documented circumstances, it's highly unlikely that you could safely and compliantly use a 250A MCB with a 300A feeder breaker and #350 kcmil conductors. A 300A MCB is the far more likely solution. Flag for review