I question the wisdom of going to the expense of converting the bike to 11-speed.
If the wheels and hubs are in good condition you could consider a switch to a 10-speed group.
Tiagra 4700 gets good reviews and will update and refresh the bike for a very reasonable price. While doing this you will have a wide range of crankset and cassette options to choose from that will fit your current frame and wheels.
I second this suggestion, or if you have some attachment to keeping the Ultegra or Dura-ace level names on your bike, then cruise ebay for NOS (New Old Stock) 9 or 10spd parts at those levels.
The actual weight savings that you will achieve by upgrading to more modern equivalents of what you have are minimal. If your goal is weight savings at a reasonable price, you will likely save more grams per dollar by upgrading your seatpost, stem, handlebar, etc...and while you are at it you would have the opportunity to dial in your position and try some of the newer handlebar bends if you are so inclined. Plus fresh bar tape of course.
If your goal is to "freshen" up the feel of your bike, newer generation parts won't help so much as just new unworn parts. Many people would argue that shimano shifting reached it's zenith with your generation of parts and that they have gone downhill since the introduction of the internal aero shifter cable routing that goes through the hood, rather than exiting right out the inside edge of the levers. Similarly, 11spd is more finicky and less tolerant of cable misadjustment or hanger misalignment than 9 or 10spd. If your parts don't have a lot of miles, it is probable that you could achieve fresh performance with just a new set of cables. If your bike has 1000s of miles on it, and the chain is sloppy and cassette worn, then add them on to the list too, along with a new small chainring. Unless you have already blown through many chains and cassettes in sloppy conditions, then I would doubt that your shifters or derailleurs have an significant wear, at least beyond what some spray lube would remedy.
But that is just my take on things. I know a lot of people like buying stuff, and are searching for ways to justify it, but since you mentioned budgetary concerns it seemed prudent to offer some alternative perspectives.