There is no single best way to do this but in my research I have come across some similar situations including for some close family as well as had people reach out to me trying to establish a connection to someone who 'may' be related to me.
It is not clear exactly what you are hoping to gain from making the connection, and in whatever approach you take I would be clear and decisive on that. Such as genealogical purposes (wanting to know), medical reasons, or just learning about 'them' as their won't be any shared experiences with those individuals. If they were conceived while in a relationship with someone else, such as their mother, that could also cause damage they do not want to deal with and a third party such as a cousin may be a better approach (see below).
In any initial contact I would make sure your intentions are clear.
A close family member of mine was in a similar situation you described and wrote several letters and eventually found a receptive sibling. Though the rest of the siblings and other family members reject the claim and want nothing to do with them. Neither has asked for genetic testing to prove it either and the relationship after initial contact is casual at best.
What you have suggested is a direct approach of having already identified the individual and crossed your T's and dotted your i's so you are definitively sure you have the correct person. A birth certificate is a lead but not definitive information in all cases (rules for what goes on a birth certificate vary by state). So part of my response is are you sure if it says Mark Johnson (made up name) you have the correct Mark Johnson, as if you approach them without definitive proof they either may blow you off and/or think you are wanting something from them.
I would strongly advise against the lawyer or anything official like a certified letter for initial contact and overall suggest starting out with a casual approach. As if you come on too strong some may even take it as you stalking them if you already know so much about them but they know nothing about you and you then may need a lawyer to address their response. If you send a couple letters over a course of several months and not sure if they are reaching them I would then maybe resort to a certified letter. You could also try social media but having to mutually following someone first to send them a message they will actually see (like on LinkedIn or Facebook or Twitter) this may not work.
A round about genealogical approach using DNA testing which can help add evidence to the claim, but also lighten the approach from the direct approach.. As one thing to consider is the birth certificate may even be wrong or have details that are not quite correct. This is a similar approach I am taking due primarily to records not being available for my grandfather who did not know his mother and the name on the birth certificate has not yet panned out yet and it is known her named changed after his birth.
So my hope is through Autosomal DNA (i.e. AncestryDNA / FamilyTreeDNA.com Family Finder / 23andMe.com) combined with genealogy document of the rest of the tree I can narrow down those VERY CLOSE matches. It adds a bit of innocent curiosity to it and if you are doing it vs. your mother may add some neutrality to it. The person you make contact with too may not end up being one of those siblings but a close cousin and they can make the initial contact on your behalf.
The catch with my round about approach is that someone in the immediate family needs to have tested and you need to understand your results as well as it takes time, something you may not have.
I also administer the DNA results for several individuals Autosomal results at this point and frequently get contacted by individuals in the same situation you described, but where they do not understand nor flush out their results and are contacting me looking for a relative or their birth parents for a 1944 birth when they are 5th-8th cousins for example with a very generic "we are related, tell me about yourself" message.
You can further refine your mother's results / matches if you and your mother get tested and then understand your results and approach your DNA matches with a message something like the following:
"I am seeking information about my mother's father, but name is not definitively known but is believed to have Mark Johnson (made up name) based on other information I found, who was a US Army Servicemen stationed in Random Military Base from June 1944 to August 1945. I see we are considered 2nd cousins and I cannot you anywhere in my family tree. Do know if this person exists in your family tree or can you tell me about any family members who might have been in this area at that time?"