Can my wife and I amend our 2019 taxes and file separately since we won't receive the stimulus check because we file jointly and I have an ITIN? She has a SSN

the IRS does not permit an amendment from joint to separate 

https://turbotax.intuit.com/tax-tips/irs-tax-return/how-to-change-your-tax-filing-status/L5yYux3qL

Also, it may turn out that filing separately creates more than an additional $1200 of tax burden.  It really depends on your situation. 96% of married couples file joint - because the tax benefits are better than filing separate.    the reason I bring this up, you could always assess this during the 2020 tax filing season next year.  Here's why:

• The stimulus payments are really based on your 2020 tax return (income, dependents, etc).
• The IRS is using either 2018 or 2019 – whatever is available  - to ESTIMATE that payment
• When you file next year there will be a “settle up” section on the tax return. 
• If you are due more stimulus (and this would include where you simply never received it), you will receive it through the 2020 tax filing.

so best to assess your tax situation next year and see if it beneficial to file separate as your spouse might be able to capture the stimulus payment, but again, you'd have to decide whether the additional taxes negate the benefit. 

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Q14. I filed a joint return with my spouse. Will we receive a Payment if I have a valid SSN and my spouse has an IRS Individual Taxpayer Identification Number (ITIN)?