Ok.... I see what's going on. If you would like to be able to use the aliases defined in .custom_rc after sh is done executing, then that won't work because the aliases are getting set in the sh session. They will die along with it when sh is finished. In other words, you should be able to use the aliases since running source up until the last line of install.sh.
You need help from yet another script that takes care of this:
wrapper.sh:
curl https://some.link/to/install.sh # download the script so you can source it
. install.sh # source it
(or something like that)
Then, you should be able to run . wrapper.sh and it should work (notice how you also source this one).
Update
About the comment about it not working:
$ cat script1.sh
alias alias1="echo hola"
. script2.sh
$ cat script2.sh
alias alias2="echo adios"
$ ./script1.sh
$ alias1
bash: alias1: command not found
$ alias2
bash: alias2: command not found
$ . script1.sh
$ alias1
hola
$ alias2
adios