The important is that you declare your revenues in the country you're operating the business. And as the url below suggests, you might also need to declare this UK bank account to the IRS. So yes, you can invoice your client in Pound Sterling and have it deposit in the UK bank account. That's a good solution if you actually have a need for the Pound Sterling somehow (eg. shopping online in the UK) and don't want to pay for the exchange rate.
But if you have an account to receive online payment such as Paypal, you can simply invoice in Pound Sterling and then convert the currency to USD, and get it transferred into your US bank account. The only downside is the exchange rate that you'll need to pay (usually around 2%-2.5% + value of currency) and the currency value on Paypal is not always exactly following the market, but it's usually close enough, and worth the "trouble".
It might be an easier and safer solution than having a deposit in the UK, and there's other online payment method you can use other than Paypal (eg. Stripe, WePay.)
Your client can either use his own Paypal account, credit card and even bank transfer in some cases. From Paypal for example, your client can deposit money from their bank account to their Paypal account, and then send you a payment.