Currently stuck on how to attach the wood pieces to achieve the X look
without having super visible joints and without compromising the
stability/support it needs.
The metal legs appear to be approximately 1" square. That is a very slender profile for wood connections. Unless you intend to apply a great deal or weight the 6" high vertical legs should be able to support the load without bowing, but the connection points to the table top and the 'X' base could be under more stress than they can handle. The problem is racking. Each connection corner acts as a pin for rotation. You want the joints to remain at 90 degrees, but any lateral force applied to the table will apply rotation force to them. Your joints must be able to resist this force and an 1" square contact will probably not be enough.
Your comment about using three pieces of wood for the x brace is also a weak solution. Gluing wood as a simple end joint is very weak and will pull apart. In addition the x brace keeps the leg bases in their proper location. To do this you want both pieces to be continuous. This means that a notched joint at their intersection is needed so that both pieces are continuous.
Work space is apartment balcony; tools miter box and hand saw.
If this is piece is intended to have any sort of 'Finished' appearance, achieving your goal with these resources will be very difficult. I would suggest you consider finding a an educational forum in your community where you could work on this project with better tools and some experienced guidance.
Finally, Making wood act like metal is a bit counter-intuitive. It is wood and performs best when the design acknowledges this fact. Based upon your dimensions it does not have to be as cumbersome as your wooden table picture above, but should be heftier than the metal base. Take a look at other images online of similar size and you should get a better appreciation for what size and styles might work for you.