BS is the entry requirement for H1B approval.

You can certainly try. Just make sure you understand the difficulty. Also read requirement 5.
___________
I-129 filing and H1B requirement

REQUIREMENT 1: Prevailing Wage - The first step in the H-1B process is to obtain a prevailing wage determination. Prevailing Wage is the amount of base salary a company must pay its foreign workers.

REQUIREMENT 2: LCA - The second step in the H-1B process is to secure the approval of a Labor Condition Application for the company. An approved LCA demonstrates to the INS that the company has agreed to comply with prevailing wage standards and other issues of employment law. An LCA must be approved for the metropolitan area in which the alien will work. If the company sends the worker to a work site different than the one listed on the original LCA, then a new LCA must be filed.

REQUIREMENT 3: Qualifications of the Company- The company must be offering a position that qualifies as a specialty occupation. A specialty occupation is one that requires the services of a holder of at least a baccalaureate degree in the relevant field. The company is responsible for providing the beneficiary with the current prevailing wage in the industry for the entire period of employment. A company may sponsor an alien for three years on an H-1B visa and may file for an extension for up to three additional years. After the six-year period expires, the alien must return to her or his home country or have made other provisions for permanent residence.

REQUIREMENT 4: Qualifications of the Beneficiary- The beneficiary must have the equivalent of at least a U.S. bachelor’s degree and must provide the attorney with a copy of her/his degree and tran. The equivalent of a bachelor’s degree is one that meets the requirements of a U.S. degree. However, the education systems in other countries vary widely, and, while some universities have four-year degrees that will equate to a U.S. degree, others do not. Many foreign bachelor degree programs require the completion of only three years. Foreign master degree programs may serve to obtain the necessary fourth year.

REQUIREMENT 5: If a person lacks the necessary equivalent degree, s/he may use a combination of her/his experience and education to meet the H-1B requirements. The standard presently accepted by the USCIS is three years of progressively responsible training/experience equals one year of university level education. If the company wishes to sponsor an alien under this premise, an experiential evaluation of the alien is required.