A hacky solution I came up with is to run a Python script that opens a window and writes the exact time to that window, then add that as a window capture to the OBS scene.
Python Script:
import tkinter as tk
import time
def update_time():
current_time = time.strftime('%H:%M:%S.') + str(int(time.time() % 1 * 1000)).zfill(3)
time_label.config(text=current_time)
time_label.after(1, update_time)
# Create the main window
window = tk.Tk()
window.title("Current Time")
window.geometry("200x50")
# Create a label to display the time
time_label = tk.Label(window, font=("Helvetica", 24))
time_label.pack(pady=10)
# Start updating the time
update_time()
# Start the main loop
window.mainloop()