[helpers] Move Base64 string to cpp

[gabest11]

What does this implement/fix?

Fix wasting and running out of memory. Also reduces fw size.

Issue #5857

Types of changes

• Bugfix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)
• Other

Test Environment

• ESP32
• ESP32 IDF
• ESP8266
• RP2040
• BK72xx
• RTL87xx

Checklist:

• The code change is tested and works locally.
• Tests have been added to verify that the new code works (under tests/ folder).

If user exposed functionality or configuration variables are added/changed:

• Documentation added/updated in esphome-docs.

[codecov-commenter]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 54.13%. Comparing base (4d8b5ed) to head (7d9f707).
Report is 672 commits behind head on dev.

Additional details and impacted files

@@            Coverage Diff             @@
##              dev    #6819      +/-   ##
==========================================
+ Coverage   53.70%   54.13%   +0.42%     
==========================================
  Files          50       50              
  Lines        9408     9619     +211     
  Branches     1654     1698      +44     
==========================================
+ Hits         5053     5207     +154     
- Misses       4056     4086      +30     
- Partials      299      326      +27     

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[gabest11]

Smart alec algo replaced BASE64_CHARS in the title with Base64 chars.