Thanks for your question. I hope I understand it correctly.
First, I assume you are using the WooCommerce plugin for your website; is that right?
You wrote “pushing products to my website“. Does that mean you are using the Products/Add New admin screen?
You wrote “product name already specified“. Does that mean you have already entered a value in the Product Title field?
You wrote “images, when uploaded“. Does that mean you are using the “Set product image” and “Add product gallery images” functions on the “Add Product” screen?
Assuming I’ve understood the above, I regret the short answer is no. If you use the “Upload” option for the “Set product image” and “Add product gallery images” functions, WooCommerce first uploads the image(s) to the Media Library and then, in a second step, associates them with the product. Thus the product name is not available during the upload process.
If you are using WooCommerce, the best alternative I can suggest is to use MLA’s Media/Assistant Bulk Edit area to update the Title values. Basically, you would select the items associated with the product(s) and apply a template to update their Title values. I can give you more specific instructions if that is an approach that would be helpful.
I will leave this topic unresolved until I hear back from you. Thanks for your interest in the plugin.