These are the reasons why you should believe a Baptist who calls himself reformed, coming from a cradle baptist who now baptizes infants.
Descriptive lexicography says that a definition of a term is set by popular use, not by decree. Whether you like it or not, the "reformed" qualifier for a subset of baptists has a function that is readily understood in context. These baptists support the doctrines of grace, the regulative principle of worship, reformed sabbatarianism, elder led polity, and other markers of reformed theology. They are distinguished from the "free will" and dispensationalist baptists, who are in fact a deviation from the historic reformed roots of the baptist movement (more below.)
An opponent to this categorization, therefore, must overwhelmingly demonstrate that the phrase itself is oxymoronic.
Most churches with congregationalist polity are not Congregationalists proper, stemming directly from the nonconformist movement in 17th c England (New and Old). Similarly, not all credobaptists are Baptists proper. Baptist is an organic stream with a certain origin. For ease, let's say the little b is the view of the particular issue (little b baptist), the big B is the movement (the Baptists), and the movement is divided into its own institutional families (Southern Baptists, American Baptists, Independent Baptists).
Commenting as a native speaker of American English, "reformed" does not involve this dichotomy. Presbyterians and Congregationalists are fairly and equally considered reformed, alongside the Dutch (e.g. RCA, CRC) and German (e.g. RCUS, E&R) reformed. "reformed" is an umbrella category, akin to the "anabaptist" label for both Mennonites and Amish. "Reformed" is also interchangeable with "Calvinist" for short, usually in reference to the so-called Five Points.
If the phrase "presbyterian and reformed" (as in NAPARC) is not taken to imply that Presbyterians are not themselves reformed in theology, then it is not a foregone conclusion that Baptists cannot be in there too.
Is this the beard of an Arminian?
There are three or four view of the origin of the Baptist movement: English Separatism, Anabaptist Kinship, and two kinds of successionism: landmarkism and perpetuity succession.
The least tenable in scholarship says that baptists have always existed in a direct and verifiable chain of churches, sometimes traced through cults, since John and Jesus in the river Jordan. It is a kind of Baptist triumphalism, and it tends to say that non-Baptists are invalid or extremely errant churches. The milder kind only says that baptists views have always been present and follow from the New Testament itself, even when the church at large neglected these issues. (Isn't this basically Protestantism as a whole?)
Anabaptist kinship alleges that the best way to think about the Baptist movement is in its relationship with the Anabaptists on the continent. This view does not require that the Baptists and the Anabaptists are the same movement, just that they are closely related. The history gets messy when 18th and 19th c works sometimes use these labels interchangeably, despite the existence of discrete movements and terms.
It is not my goal here to persuade the reader, but the English Separatist movement is the most appropriate theory of origins. The Baptists are in short credobaptist congregationalists. Think only about the predominant Baptist confession, the Second London. Westminster was drafted by the Assembly, for the Presbyterian movement. The Congregationalists in London adapted it for their movement (Savoy). The Baptists then adapted Savoy for themselves. Hercules Collins did the same with the Heidelberg Catechism, and it should be admitted that the original First London Baptist Confession shared the same distinctive theology, as does the Abstract of Principles for Southern Seminary in the US.
Did the confession become non-reformed? Not by mere principle of alteration, because the Congregationalists did the same without opposition. The opponent to the phrase "reformed baptist" needs to demonstrate the single issue of credobaptism breaks the "reformed" designation, and not only that it is idiosyncratic or false. Careful Baptist theologians are willing to admit that the change in the sign of the covenant has other implications for covenant theology.
Think also about the movement in the US: Roger Williams obviously moved out of a Reformed, Congregationalist context in New England. "But", I hear you say, "So did the Mormons schism from a Reformed context, and so did the Remonstrants!" Yes, and the burden of proof is on you, inner voice, to demonstrate that the degree of change from paedobaptist Congregationalist to credobaptist congregationalist throws out the Reformed moniker. The Arminians were rebuked at the Synod of Dort. The Mormons are restorationists - their central argument is that the church of all ages was false or absent until they dug up some new scriptures and then translated them by staring into a hat (look it up).
Can the doubters prove their point? I doubt it. The semantic range of given terms are always in flux. Consider the extra wrinkle that even the Heidelberg Catechism and its original proponents were not officially, technically, Reformed. They claimed to stay within the bounds of the Augsburg Confession during the Peace of Augsburg, and they were right. The Reformed movement, like any other, had its origins, a period of formation, then a settling long after. Would "reformed baptist" be an anachronism in Germany in 1563? Yeah, probably. I'm only suggesting that we stop interrupting the Baptists to say "Um, Actually!" when they bring up Calvin.