Yes, B can file suit before the Bombay High Court in the instant case.
In Union of India v Tantia Construction Pvt. Ltd, it was held that the existence of an arbitration clause does not itself bar the writ jurisdiction of the High Courts and the Supreme Court under Articles 226 and 32 of the Constitution respectively. Furthermore, in Bhatia International and Venture Global Engineering the Supreme Court had held that Indian Courts could interfere to the fullest extent allowed by Part I of the Arbitration and Conciliation Act, 1996 (Act) even when the seat of arbitration was outside India. (The New York International Arbitration Center is the seat of arbitration.)
A recent case involving Reliance Industries Limited and BG Exploration and Production India as the parties had similar facts, and the SC ruled that any final arbitral award can be challenged only in the English courts, but substantive Indian arbitration laws will have to be applied by the foreign courts. The English courts will also observe the India public policy while deciding the issues.
Courts in India have jurisdiction to try all suits of a civil nature excepting suits of which cognizance is either expressly or impliedly barred . This means that the Court cannot entertain the suit if there is a referral to the arbitration clause directly or indirectly barring the jurisdiction.
Foreign awards are enforceable in India , treated as decrees of an Indian Court. It has further been held by the Supreme Court that, under the Arbitration Act, there is no requirement for a separate decree to be passed by a Court of competent jurisdiction for the foreign award to be enforced. All other foreign awards will need to be ratified by the courts and a decree passed by the court, which will then become enforceable.
Indian courts do recognize and enforce a choice of law and jurisdiction made by parties as long as the decision is being made in good faith and not against public policy. However, Indian courts may refuse to enforce a condition regarding the choice of forum where it is of the opinion that such choice is oppressive, unfair or inequitable and does not bear any real or substantial connection to the subject matter of the dispute. The Court also takes into account the balance of convenience, and interests of justice for upholding the exclusive jurisdiction clause of an agreement.
http://lexarbitri.blogspot.in/2011/05/arbitration-clause-does-not-bar.html last accessed September 7, 2015
last accessed September 6, 2014
Section 9 of the CPC, 1902 (Civil Procedure Code)
Under the New York Convention on the Recognition and Enforcement of Foreign Arbitral Awards and under the Geneva Protocol on Arbitration Clauses
http://khemkafoundation.in/index.php?option=com_k2&view=itemlist&task=category&id=49:dispute-resolution-in-india&Itemid=283 last accessed September 6, 2014