jq and bash: object construction with --arg is not working - bash

Question

Given the following input:
J='{"a":1,"b":10,"c":100}
{"a":2,"b":20,"c":200}
{"a":3,"b":30,"c":300}'
The command
SELECT='a,b'; echo $J | jq -c -s --arg P1 $SELECT '.[]|{a,b}'
produces
{"a":1,"b":10}
{"a":2,"b":20}
{"a":3,"b":30}
but this command produces unexpected results:
SELECT='a,b'; echo $J | jq -c -s --arg P1 $SELECT '.[]|{$P1}'
{"P1":"a,b"}
{"P1":"a,b"}
{"P1":"a,b"}
How does one get jq to treat an arg string literally?
Using tostring gives an error
SELECT='a,b'; echo $J | jq -c -s --arg P1 $SELECT '.[]|{$P1|tostring}'
jq: error: syntax error, unexpected '|', expecting '}' (Unix shell quoting
issues?) at <top-level>, line 1:
.[]|{$SELECT|tostring}
jq: 1 compile error
SELECT needs to be a variable and not hardcoded in the script.

Answer 1

SELECT needs to be a variable and not hardcoded in the script.
Assuming you want to avoid the risks of "code injection" and that you want the shell variable SELECT to be a simple string such as "a,b", then consider this reduce-free solution along the lines you were attempting:
J='{"a":1,"b":10,"c":100}'
SELECT='a,b'
echo "$J" |
jq -c --arg P1 "$SELECT" '
. as $in | $P1 | split(",") | map( {(.): $in[.]} ) | add'
Output:
{"a":1,"b":10}

Answer 2

If you really want your data to be parsed as syntax...
This is not an appropriate use case for --arg. Instead, substitute into the code:
select='a,b'; jq -c -s '.[]|{'"$select"'}' <<<"$j"
Note that this has all the usual caveats of code injection: If the input is uncontrolled, the output (or other behavior of the script, particularly if jq gains more capable I/O features in the future) should be considered likewise.
If you want to split the literal string into a list of keys...
Here, we take your select_str (of the form a,b), and generate a map: {'a': 'a', 'b': 'b'}; then, we can break each data item into entries, select only the items in the map, and there's our output.
jq --arg select_str "$select" '
($select_str
| split(",")
| reduce .[] as $item ({}; .[$item]=$item)) as $select_map
| with_entries(select($select_map[.key]))' <<<"$j"