Wow! I'm actually quite surprised you knew about her. Though she is a famous one in the last days of Ming Dynasty, she wasn't very well-known amongst the wide array of skilled and respected generals throughout the last 5000 years of Chinese history.

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is it possible she would have been awarded even higher duties if she were a man?

Possible at certain point, but not very likely by the end of her career. I will look at it by going through briefly her whole career as a great general of her time.

Her first appointment was from her late husband, Ma Qiancheng, who was a 宣抚使 (Xuan Fu Shi -- something like a city/region commander) at present day Chong Qing. After her husband died in prison (he was imprisoned due to framing from other officials), Liangyu took over^(1) the appointment, which is a 从四品^(2) (Cong Si/4 pin). This happened in 1613 .

In laymen terms for appointment ranking, the smaller the number, the higher the ranking. So 一品>四品 (1>4). And within the same number, 从 (cong) is smaller than 正 (zheng). So 正四品>从四品.

In 1620, The Later Jin launched attack on the Ming borders, Liangyu was deployed to help in the frontline defense, and promoted by the court to become a 三品^(3) (san/3 pin).

In 1621, after her brother died in a tough fight in an attempt to break siege. Liangyu led her troops to support the remaining soldiers who managed to survive, and to aid in the defense of 山海关. The emperor of the time, Tianqi, promoted her to 二品 (er/2 pin) and bestowed her the title of 诰命夫人^(4 5) (Gao Ming Fu Ren = Mrs xx by imperial mandate) . 诰命夫人 is a prestigious honorary title given to an official's wife or mother , as an recognition and reward for his/her tremendous contribution to the empire. It's ranked from 一品 （yi/1 pin) to 五品 (wu/5 pin).

So how did the court in general react to her rapid promotions and achievements? Would it have hindered her further promotion to higher duties and titles? Possibly.

臣率翼明、拱明提兵裹粮，累奏红崖墩诸捷。乃行间诸将，未睹贼面，攘臂夸张，及乎对垒，闻风先遁。败于贼者，唯恐人之胜；怯于贼者，唯恐人之强。如总兵李维新，渡河一战，败衄归营，反闭门拒臣，不容一见。以六尺躯须眉男子，忌一巾帼妇人，静夜思之，亦当愧死。

This is an excerpt from her personal biography^(6) (written by historians slightly after her time). It was a letter she wrote in 1623 to the emperor about the mistreatments she received. It's rather long so I will not do a word-for-word translation, here's the rough one ' I fought brave and hard for the empire, but there are people spreading disharmony and distrust between you (the emperor) and me, and who has never even seen the face of the enemy but dare to boast their power and might. Then when the real enemy comes, they are the same people that will ran in fear without a fight. Such as one official, who after being defeated, went back to the encampment and refused to let me in upon my return. For such a strong and mighty man to be envious and jealous of a mere woman, he should be ashamed,'

This provides some insights that there may indeed be people , be it within the court or her fellow generals, who were envious of her achievements and promotions. And went on to find ways to bring inconveniences for her or even bad-mouthing her in front of the emperor. Especially since the imperial court was far from peaceful. Tianqi's reign was plagued with dominant and powerful eunuchs and their allies who will relentlessly destroy their competitor's career or even hunt them down should they feel the need to, (E.g. 魏忠贤) as well as constant conflicts in court between 东林党 and the aforementioned group of eunuchs.

However, not all court officials were malevolent. The Minister of Defense, Zhang Heming, actually advocated for the recognition of her achievements and better welfare of her family back in 1621 following her brother's death^(7).

Moving on to after the death of Tianqi and the reign of the new emperor, Chongzhen. In 1630. the Qing army lead by Hong Taiji advances to the Ming capital, Beijing. Liangyu was summoned to aid in the defense , this was also when she met the emperor and was given the poems you mentioned^(8). There were also sources claiming that she was promoted to 一品（yi/1 pin) 诰命夫人 here but credibility seems questionable (it wasn't mentioned in the official biography that I was referring to most of the time) so I did not include it here.

In the following last years of the Ming Dynasty, Liangyu was frequently involved in neutralizing revolts across China, but as the revolts get stronger and more organized (the people were increasingly unhappy), there's nothing much that she can do, especially since the Ming's official grain reserves are running low across all the states due to constant revolts.

In 1644, the last year of Ming, Liangyu fought with revolt leader Zhang Xianzhong, but was defeated as she was outnumbered. She declared that she is a woman of Ming and will fight till death for Ming, never surrendering to the likes of revolts like Xianzhong.^(9) As such, she retreated back to Chong Qing, Xianzhong went on to convince and persuade neighboring towns and cities to surrender, but dare not go close to Chong Qing in fear of her.^(10)

That marks the end of her career with Ming Dynasty. Looking at it as a whole, she eventually attained recognition and title that even many man were unable to, she was also entrusted to significant duties like defending borders and capitals and neutralizing revolts. With that, as well as the chaotic state of Ming in the last few years of her career, in mind, I would say that she has more or less been awarded pretty much all of the highest duties and recognition possible , besides gaining a marquess or similar title that is. (Which I will briefly touch on later)

Her story doesn't end here though. In the following years that Qing Dynasty was trying to unify China, there exist numerous smaller self-declared successors of Ming. Most prominently the Southern Ming in South China. In 1646, when she was 73 years old, the Yongli emperor in Southwest China granted her the title 太子太傅 (Tai Zi Tai Fu =teacher of the crowned prince). the Longwu emperor in Southeast China also granted her a similar title 太子太保 (Tai Zi Tai Bao) and the title of a marquess^(11) ---- 忠贞侯 (Marquess of Zhong Zhen = Marquess of loyalty and faithfulness).

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Or did the Ming court typically not promote regional "warlords".

I believe you mean marquess or any similar kind of noble titles. Warlords were mostly not allowed and encouraged in most of ancient China's unified dynasties. (because it signifies fragmentation and warlord's people/armies often only follow their warlord and not the emperor. None of the emperors like that idea) . Well, the evolution of giving out of these titles is rather complex in Ming. Go into that and you will never finish reading this reply. In summary, yes, Ming courts, especially late Ming periods, tend not to promote military -related personnel to such titles. This could be partly due to the array of problems and their consequences during early Ming periods when giving out such titles , such as Jingnan Rebellion, which eventually resulted in the overthrowing of the rightful crowned prince of the previous emperor and hence the rightful current emperor by his uncle, or the Gaoxu Rebellion, just to list a few.