DISCLAIMER: Don't do stupid shit, consult with a professional.
Ok, so I'm not going to tell you what to do because there are a lot of variables, speak with a licensed electrician. You don't have to upgrade to a 200amp service for something so simple, there are other options. Not saying that you shouldn't upgrade, depends on your circumstances, but there is no way I would spend $1000-2000 to have a service upgrade done for that.
I will say that here are some things that have been done in the past that might work for you

Remember, distance is your enemy(esp with undersized wires), and your garage wiring adds x feet to the wiring, increasing resistance.
1. put a damn 20amp circuit in there, there are breakers that have TWO circuits in the space of one, leaving you with an extra slot to install another circuit, or remove one other circuit that isn't being used, or pigtail off of one circuit breaker for two of your light circuits(speak to an electrician about which ones can be combined), or, or, or... plenty of options. Yes splicing/pigtails are allowed inside your loadcenter.
2. do you have an electric dryer circuit that can be used? Oven circuit? You can use an adapter like this example:
https://amzn.to/2YgIS0Y
3. use a large generator.
I'm sure there are more that I'm not thinking of at the moment, but you definitely don't need to upgrade your service for a damn welder. ESP if you add a 'welder only' circuit.