Lunging pads and cavessons
If you don't want to lung your horse using a bridle or halter you can choose a cavesson. This is basically a sturdy halter with three rings on the noseband to attach the lunging rein or reins to. Another useful tool for lunging is lunge coupling. A small strap that can be attached to the bridle's bit rings; the lunge rein in turn is attached to the lunge coupling. Lunge couplings are available in many fun colours and motifs by brands like Imperial Riding and Epplejeck.
If you want to protect your horse's back from chafing and irritation from the surcingle you can use a lunging pad to help distribute pressure. Lunging pads are also available in several fun colours, like the Friesian lunge pad by HB.