Yes, you can make the first year choice to become a resident alien starting July 23. You may file an extension for your return and file it at the time you satisfy the Substantial Presence Test requirements. it seems like you have a good understanding of the requirement in your calculation.
You may claim your wife as a resident and you may file a W7, an application for an ITIN. The caveat to this is that she needs to sign the application which could be problematic if she is overseas. The only thing I can advise is to get this process started immediately if you wish for her to have an ITIN. Please read this IRS publication for instructions to file the W7 and get this process started without delay.
Please read the procedure carefully for making the first year choice so you will be prepared when you are ready to file when you meet the conditions of the Substantial Presence Test.
First Year Choice
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