Ask your structural engineer.

@PPF. Below are the pictures:

Current Beam:

Proposed bump-out:

The issue isn’t the structural. It’s the kitchen design around the structural. As shown, it is not an acceptable design response to the structural issue. You cannot ignore the required intrusion into the space, which is what they are trying to do. It must make sense within the whole. That is not a good location for the sink as shown. Your structural engineer may be well qualified, but your kitchen designer is lacking. When you’re separating two spaces that still will be connected, you have to treat them as 2 separate but connected spaces.

Your structural issues sound like they will be realistically realized cost effectively. It's not all that. Find a good framer that knows an engineer please.

Fall back to square 1, as things exist now. You need a different design.

Of course you should speak to a structural engineer, but my sense is that it should be possible to do a flush beam over such a short beam span, and that the beam could be terminated inside the exterior wall, with no protruding support post. We usually do flush beams that cover much greater lengths than this (up to 30'), and if an external wall is at one (or both) ends of the beam, we always bury it in the wall.

You didn't mention the supported length (the average length of the joists bearing on the beam on both sides). That could have a bearing on how large the beam needs to be.

You need to provide a lot mor einfo and you need a structural engineer to plan this and be prepared to spend $$$$ to do it even if possible . Since you only have 10” you will need to know how big the beam needs to be and I bet steel will be the answer. BTW the pictures are no help at all.

Sounds doable but work with your structural engineer. In my last house (13 steel glitch plates and point loads through a four story building to the foundation) the engineer provided drawings, I asked for some aesthetic tweaks (switched one from wood to steel to make it smaller) then we headed for permits.

Have you done exploratory demo to see what’s actually in the wall?

I don’t know about the structural, but I do know you can’t have a shallow wine cooler like that, in front of that post. You have to have a 24” deep space for it. I had a shallow area that I wanted to do a wine bar. There aren’t any 18” deep refrigerators or coolers. My fabulous kitchen designer finally figured out to put one in sideways!

Thanks for all the feedback! The information I am getting currently is just from my designer, drawings will be in an engineer's hand next week and I will be talking to him personally. Just wanted some high level insight from the wonderful folks here, to know potentially what I was dealing with. We also very much agree with most all of you that the pictured "solution" above that was proposed, is not at all a solution we are willing to settle with.