After you file your amended return, you can absolutely make the request for the installment agreement.
I wouldn't suggest sending the 1040-V with the installment agreement request (Form 9465) as you may generate some confusion when the IRS is trying to process it.
I actually recommend filing your Installment Agreement Request to pay your amended balance due online at https://www.irs.gov/payments/online-payment-agreement-application
The IRS charges an application fee for payment plans - the fee is $31 if you file online, and $107 if you file your payment plan request by any other means (by mail, phone, etc.) Filing the application online saves you some money (that you can put towards your balance due, which will help minimize the interest you end up paying on the taxes you owe.)