Just passing through, I'm not an academic and can't speak to the specifics of academia. But I am someone with an interest in the legal use of pseudonyms.
The legalities of using a name other than the one on your birth certificate vary, obviously, by legal jurisdiction. In the USA, there are some Federal issues, but substantially it's up the the individual states, and their attitudes and procedures around this vary widely.
There are two populations that care very much about this issue: trans people and sole proprietors (people in business for themselves, who have not incorporated). This can be an excellent and informative lead for finding information specific to a jurisdiction: looking into how both those populations go about changing their names or working/functioning under something not (yet, or fully) their legal name.
For instance, I was recently surprised to learn that California has extremely liberal rules for name change, to the effect of, so long as you're not deceiving people as to your identity to commit crimes, you are whoever you say you are. Legally. This is called "name change by usage". Other states might consider that fraud.
Meanwhile, most(?) states have some way for people to officially register a name under which they do business. Sometimes this is called a "doing business as" or "DBA". This is what very small business owners do, when they are a "sole proprietor" (legal, IRS term, which means they are their business) and want to have a business name. Note, this is entirely different from incorporation. It merely creates a public record that, say, "Joe Smith" is "dba" (doing business as) "Joe's Plumbing" or "Frantabulous Widdershins, Jr." I gather many states do this on the county level, but here in Massachusetts, it's handled by municipalities (and BOY did this confuse the computer at my new out-of-state bank), where it's called a "Business Certificate". You go to your city hall, fill out a form (which, note, makes your new business name, its connection to your old business name, and your home address all a public record together), and pay them some money (I paid I think $60 for a four year certificate) and, boom, you're good to go.
Why do this? So that you can, say, walk into a bank and open a bank account with that name, so that people who want to pay you by writing checks can write them to the name you prefer and you can still deposit them - something all the banks I talked to would not do without seeing the official embossed Business Certificate. I don't know to what extent payment processors like PayPal care about matching names on bank accounts and credit cards when they're being linked to the account, but that may also be an issue.
I found out that there are relatively recent Federal laws for renting mail boxes at mail box services (PO Boxes, only from commercial services) that require them to have your "real name" - again, my Business Certificate made it acceptable (and maybe legal?) for me to have my business name on my box (and receive mail for my business!) instead of just my personal legal name, but it also requires my legal name.
There are a whole bunch of places in life where we do business with our names, and it might be sticky or awkward for you to be functioning under a name not on your ID. Going out drinking with colleagues and getting carded. Getting your ID checked for a flight to a conference (being paged by the airline to come to the white courtesy phone). Reserving a hotel room at a conference, which has to be secured with a credit card. Splitting the cost of a pizza with Venmo.
On top of all that, there have been some huge controversies about social media giants – Facebook, Google – having "real name" policies which lead to users, including some quite famous ones, getting locked out of their accounts for using pen names, stage names, etc. instead of their "real" names. Facebook kicked out Salman Rushdie, of all people – one of the most famous novelists in the world – because his legal first name is "Ahmed". Basically, using any sort of pseudonym opens you up to the possibility of this kind of bureaucratic harassment. It helps to have some legal paperwork – but it might not be enough.
I would caution you that using a pseudonym – even a very open one (the connection between nym and legal name is not at all a secret) – is generally, slowly becoming more and more difficult in the US and treated more and more prejudicially. Fully-fledged legal name changes don't have that problem. But increasingly, systems – business, legal, social, technological – treat people who try to function under more than one name as suspicious and possibly criminal.
Some of that is ignorance, and hostility to the notion of people being "two-faced". But I also have been entertaining the hypothesis that having multiple names is a kind of class marker. The populations which have multiple personal names tend to be those with reputational management issues - which you, as an aspiring academic, can appreciate. Those issues can be lumped into two rough piles. For one, there are people managing stigmatized identities, such as immigrants who Anglicize their names when in the workplace, but continue to be known as their original, traditional names among friends and families. For another, there are people whose identities are "brands", such as in your case. Most of the individuals whose identities are brands they need to manage for the sake of their career success? Are usually in some line of work which is either intellectual or artistic or both.
There are plenty of people in the US who are hostile to immigrants and intellectuals, and I think some of them take finding someone has two names as indicative they are likely one of them – and that's not even erroneous – those outsiders and "elites" who do weird things like have two names.
Wherever it comes from, public policy and private attitudes in the US are slowly swinging towards insisting that people have one, official name, and use it for everything. Expect more obstacles to crop up, over time, if you try to function under a pseudonym that doesn't have some legal underpinning, and maybe even if it does.
This may not be a problem if you're comfortable treating your professional name as a kind of nickname, and always fall back on what's in your wallet. But it will be a thing that comes up. Possibly more than you imagine. Possibly more than I imagine.