Thunderbird can't do it. That is one of the e-mail client/editors I have tried. It only allows very limited editing capabilities as far as I can tell. It allows you to add a table, but I don't see any way to add a background image for the table. It allows you to add a background color and you can add attributes. I tried adding a div to see if I could use that to insert the image, but there is no way to edit the div after it is created as far as I see. There is no way to edit the code in the e-mail directly that I know of.
I created code that displays what I need on a web page, but if I paste the code or table displayed on the web page (stored locally on my computer), then the image in the background will not be displayed in the e-mail online. If I upload the image online and use an external code, there is a risk of legal consequences because it looks like a stock image.
Gmail has something called google drive that says you can store files on your e-mail (basically like an attachment, but with files stored on your e-mail, not sure what the difference is), but I don't see a way I can get the html code for the table to link to any images attached using google drive or other method to display images in the e-mail. I'm not sure if there is a way to embed an image displayed for the background of a table or div tag in an e-mail.