Spend less on the video capture device, spend more on lighting.
"Time lapse" is a way to speed the passage of time on a given subject. There are two ways to do this:
1) Take a bunch of stills at regular intervals. Using a video editor that can deal with these stills, place the still images in order on the video editor timeline. Understand frame rate in combination with the duration interval between image captures. Adjusting the amount of time the still image stays as the video along with the required playback time of the sequence... finally, render the resulting video from the video editor. Frame rate can range from 15 to 30 or more frames per second. Capture can vary... taking an image every 5 minutes, hourly, daily... it depends on the sculpture and how big the changes are from image to image.
Tip: The image capture device should stay in the same place for the duration of the project. Moving it is possible, but the image capture duration should be considered - at worst, move the image capture device very little - perhaps after about 1/3 of the time of the duration until the project is planned to be finished.
In this case, I don't know if a webcam can be optioned to capture still images at a preset interval.
2) Capture the entire project to video, import that video to a video editor with a "speed" function, increasing the playback of the segments to where ever you want it. I did a time lapse activity of an eight hour drive from Southern California to Northern California. After editing (including speed-up), the final rendered video was ~6 minutes. The camcorder was mounted to a tripod and in the fronts seat of the vehicle... The vehicle moved - the camcorder did not.
Since we don't know what the lighting environment will be during the sculpture work (outside under sunlight; inside under poor or other lighting), we don't know if additional light is needed. In theory, any entry level camcorder (I suggest Sony, Panasonic, Canon, JVC) will meet your HD recording and "selfie style" (flip the LCD display) technical requirements. As well, you *should* be able to connect the camcorder's video-out to a monitor - this could be a 7 inch field video monitor or a 65 inch flat panel TV. This is not possible on all entry level camcorders, so be careful. They won't record directly to a computer. They will need to be able to capture video which plugged into an external power source (the wall AC outlet) - sometimes, when the external power source connects to the camcorder's USB port for recharging the battery, the camcorder cannot record... again, be careful. Unless you buy previously owned, they will cost 4x to 7x more than your $50 budget.
For speeding up the video... Whether you do a lot or a little video editing, step 1 is to get the captured video into a video editor. Note that when the video is imported to the video editor, 60 minutes of High Definition 1080p video will consume about 44 gig of computer hard drive space. It is recommended that the video project files be on an external drive connected to the computer (for "best" response time from the computer during importing, editing and rendering).
When you decide on a device, don't buy it. Go to the manufacturer's website, download the manual and read through it. If the information you want is not in there, then ask the retailer or manufacturer. No info on a specific capability may mean that specific capability is not supported so you will need to drop the device and move to the next one. When all your requirements are confirmed, then buy it - and if the capability fails to appear, you can return it with the reason that "you said it could do ____, but it does not"...