Rewrite Template Management Client based on Pydantic Models

## Description

Rewrite template management client to use Pydantic models from the API.

## Tasks

- [ ] Generate client from Pydantic API models
- [ ] Implement type-safe template methods
- [ ] Add client-side validation
- [ ] Update client tests
- [ ] Update client documentation

## Acceptance Criteria

- Type-safe client interface
- Full IDE auto-completion
- All tests pass

## Dependencies

- Blocked by: BA-2635 (Migrate Template Management API to Pydantic)

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JIRA Issue: BA-2645