There are a variety of ways to print on printed silk fabric. Celeste is right that screen printing would likely just affect one side of the garment. But you could certainly screen print both sides.
She’s also correct that some screen printing ink (especially the kind used on the majority of mass-produced tees, plastisol) would not have the soft feel of a dye. But screen printing is not limited to plastisol. You can screen print water-based ink or dye to achieve a soft hand.
The biggest issue I see is getting the image of what you want printed. You do need to get a high-res scan of the artwork. So you would paint a “model scarf” by hand which would be used to create the future iterations. Once the art is scanned it is possible to separate colors and create screens. Based on the specific picture you sent, you could use just two screens (gold and purple) to achieve the print. You could even mimic the dark areas where the dye overlaps. That would require testing and using ink or dye with an appropriate amount of transparency. You would also have the option to use different ink/dye colors to create different colorways based off the same art/screens.
Celeste also brings up a good point about print size. Most t-shirt printers are not set-up for larger-format printing. That being said, larger print sizes are not prohibitive. Windshields, snowboards, and wall paper get screen printed daily. In my studio large table napkins, table runners, and pillow-cases are screen printed.
Unfortunately dye sublimation is not an option because it only works on polyester fabric. The only other viable option is to digitally print on printed silk fabric. Spoonflower (Products & Pricing – Spoonflower) is a consumer-facing business that does just that. Without running the numbers, my guess is that the cost is going to be higher than screen printing at volume and the colors will not be as vibrant. But it’s certainly an option to consider, especially if you’re testing designs or a concept.
printing on printed silk fabric is one-sided, but one printed silk fabric which is printed on both sides is printed silk fabric chiffon (which I often hand-paint for printed silk fabric scarves and clothing).
The printed silk fabric chiffon is much finer than the other types of printed silk fabric, and companies who do printed silk fabric printing can generally print printed silk fabric chiffon with the design visible on both sides (as the printed silk fabric as almost see-through).
I highly recommend you search an online marketplace like alibaba dot com and search for printed silk fabric printing — you’ll find a whole host of companies who do mass production of printed printed silk fabric by the metre, some who will print AND turn the printed silk fabric into scarves etc. The companies might be in China, India etc — make plenty of enquiries and ask about the types of printed silk fabric where the design will be seen on both sides, and ask for small quantities or a sample first before agreeing to any quantities.