How to best arc image without distortion to fit curved cup template?

I have tried searching forum for this and I know it has been discussed before but not really finding what I need to do correctly.  I have attached the layered image (reducing opacity so you can both template and image). The vendor provided me a flat template, not a path, to do my artwork. As you can see I need to arc the image without distorting to fit a cup template. I am more comfortable in photoshop but I can export to AI if necessary to complete which appears what I need to do via envelope distort from what I keep seeing. However, the template provided is not a path. I tried to trace template shape manually with pen tool and use envelope distort to arc my image to conform but it looked very distorted and no even close to what I was trying to achieve. I tried to do tradition arc warp in photoshop but it still seems to distort proportions. Steps on how best to do this would be much appreciated! I know there must be a easier/more efficient way to achieve this.

I would not trace the template and will simply use it as a visual reference for deforming (conforming) the image to it. You can make the template image transparent to certain degree using the Transparency panel so you can see through it then move its layer on top of all other layers and lock it. Scale and position your image in such way that at least two of its corners fit two corners of the template for example the two top corners and then its bottom edge is on the same level as the template.Then select the image to be deformed and in the control panel at the top of your screen click the Embed button which allows you to deform the image (edit: just noticed you wrote it is already embedded, so skip this).
Choose Object > Envelope Distort > Make it with a Mesh. and chose 1 for rows and columns. Make sure the Smart Guides are on (View > Smart Guides). With the Selection tool (white arrow) Shift click or drag a selection box to select the two points of one of the vertical sides of your image. Pick the Sheer Tool and first just click on the selected top corner point to set the reference point of the sheer then click and drag the corresponding bottom point to match it to the corresponding point of your template. You may hold Shift while dragging to constrain although the Smart Guides will do that by snapping to edge of the envelope if your mouse is close to it. Repeat this for the other side of image. To make the image curved, pick the Selection tool (white arrow) click a corner point which will display its handles, pick a handle that is on a horizontal side of the image and pull it up to deform the shape referring to the template. You may do this first roughly with all horizontal handles and then you can drag guides from the ruler when a handle is selected to snap a guide to it and use it to put the corresponding handle on the opposite side on the same level.
Once you have deformed the image to conform to the template, you can fine tune the distribution of the vertical deformation with the vertical handles of the corner points. You can grab a handle and drag it along the vertical paths of the envelope (the Smart guides will snap to it). Drag the vertical handles on one each side lower until you like the distribution of the deformation.
Try to follow my instructions and I'll clarify if you are stuck.

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