Yes, it is possible to interact with instances of objects with the mouse even though they have no sprite with a bounding box assigned. You're not going to get around "a lot of work with coordinates", but I wouldn't exactly define it as "a lot".
You can determine the boundaries of text being drawn by keeping an instance's coordinates (x, y) and the current horizontal and vertical text alignment (whatever was set via draw_set_halign/valign - there is no way to retrieve this as of writing, so you'll have to assume the same settings you use when drawing) in mind and using the functions string_width and string_height. To quote the manual, they calculate "(the width and height of) the string as it would be drawn with draw_text using the currently defined font".
Using all of this data, you should be capable of constructing a rectangle which outlines the text being drawn (essentially like a sprite's bounding box) and can then compare mouse cursor coordinates to check whether it's inside the rectangle.