For answering this kind of question, I would like to tell about history for easier understanding at first.
In the beginning, there was indigenous Japanese language which is now called as 和語 or やまとことば. They didn't have letter or writing system.
Since Chinese language was imported, 漢字 were used for writing 和語 but pronunciation were maintained. At the same time, using 漢字 with its Chinese pronunciation also started. Moreover, even ひらがな and カタカナ were invented from 漢字.
Therefore, most of 漢字 has both 訓読み as pronunciation from 和語 and 音読み as Chinese pronunciation. Some 漢字 have several 訓読み because same 漢字 was assigned to different 和語 word but also different 漢字 were used for same 和語 word as well.
The character 支 was assigned to different 和語 of ささえる and つかえる so that 支える can be read in two ways and certainly in different meanings.
差し支える is さしつかえる. This [支]{つか}える has different meaning from [支]{ささ}える, support, maintain, hold....
[支]{つか}える means "to be blocked" "to reach a dead end" "unable to go ahead because of blocked/clogged way".
- 天上に頭が[支]{つか}える。
- 餅が喉に[支]{つか}えた。
- 早くしてくれ。後ろが[支]{つか}えている。
- つっかい棒
Now, about 差し.
This word was derived from verb 差す but since archaic Japanese, this has been used as prefix for emphasizing another verb.
- 差し迫る
- 差し止める
- 差し挟む
- 差し引く
- 差し控える
So, 差し支える can be said that it is simply of same meaning with [支]{つか}える.
There is no 差し[支]{ささ}える in authentic Japanese word vocabulary.