To get back to the specific question in the OP,
this is why the President must not compromise, aka give in to GOP demands, under these circumstances:
1. The President cannot not let the holding hostage of government funding become a legitimate, workable bargaining strategy in budget negotiations.
2. If the radical faction in the House succeeds with the above strategy, their lesson learned will be that this is the new way to impose their radical agenda on the rest of the country.
3. This would never end. Here is one list of everything else the GOP has demanded on threat of shutting down the government:
1. A balanced budget amendment [Link]
2. Approving Keystone XL [Link]
3. Eliminating funding for Planned Parenthood [Link]
4. Medicare privatization [Link]
5. Tax reform, as outlined by Paul Ryan [Link]
6. The REINS Act, which would require Congress to approve significant federal regulations [Link]
7. Means-testing Social Security [Link]
8. Defunding Obamacare [Link]
9. Allowing employers to eliminate insurance coverage for birth control [Link]
10. An expansion of off-shore drilling [Link]
11. Preserving all the Bush tax cuts [Link]
12. “Trillions” in budget cuts [Link]
13. Slashing funding for food stamps [Link]
14. Protecting mountaintop strip mining [Link]
15. Stripping the EPA of authority to regulate greenhouse gases [Link]
16. Loosening regulation on coal ash [Link
21 Things Republicans Have Demanded To Avoid A Crisis