The crack you describe sounds like its by a lead “wiped ” joint VERY EASY to repair in most cases and can last many more years if the base metal is intact.
To repair this kind of crack you have to clean this area with a fine wire brush or shave hook, being very careful to remove all oxidation completely on both the lead base metal and solder.
Now you can do like I do and use a ladle with Flemco # 1 Solder (40-60) and lead wipe a solder patch over this crack using sperm candle as a fluxing agent.
Like your job for example it is very cost effective to get a LMP to do this kind of long lasting repair.
I did this kind of repair in a Museum where the management honestly did not want the inlaid wood ceiling destroyed and opted for the price I gave them to do this 1/2 hour job.
If your not up to lead wiping with a ladle and cloth you can use a mixture of 38-62 solder which has a longer pasty range and you can use a soldering copper (Irons) Or
you can use a standard soldering tip NON TURBO TORCH TYPE and cheat the sperm candle by use a self tinning solder /flux like Hercules
Or you can use one of the many self cleaning fluxes out on the market with spool solder like 40-60 0r 50-50
Any semi skilled journeyman can solder Or lead wipe a lead pipe.
A lot of stumblebums have no consideration for their victims and say “replace” rather then admit they are non skilled in the trades and look for an easy out way of doing these jobs.
Now for point of information If you can find a local “plumber” who will make this permanent type of joint don’t be shocked by the estimate he/she gives you.
I charge a minimum of $1,000 BUT it is well worth it considering the alternative of tearing down ceilings and repairs and the having the work completed in the SAME DAY
I am sure you can find a local guy willing to give away his talent or you can try some of the newer types of epoxies that the new generation of apprentice is fond of using.
Lead piping lasts and lasts for well over 100 years if maintained properly, many of my older building accounts still have not only lead waste piping and vents BUT lead water main piping.
Have a licensed plumber check the other connections while you have this opening to make sure this repair is warranted.
The Master Plumber may also opt to cutout a corroded section of the drainage lead “D” lead and inset copper tubing inside as a “splice piece” and either solder Or lead wipe each end to make sure he/she has water tight integrity.
The key is to find a skilled craftman GOOD LUCK